3: Mechanical Waves(Chapter 16)

Phys130, A01Dr. Robert MacDonald

Mechanical WavesA mechanical wave is a travelling disturbance in a medium (like water, string, earth, Slinky, etc).

Move some part of the medium out of equilibrium, and that motion travels (or propagates) from one place in the medium to another.

• Since it took energy to disturb the medium from equlibrium, that energy propagates with the disturbance. (Different parts of the medium are moving as the wave moves; that’s kinetic energy.)

Waves on a string are a good basic model to use to explore this.

2

Waves: Space and Time

0 2 4 6 8 10

x (m)

-2

-1

0

1

2

De

lta

y (

mm

)

2 m/s

t = 0

Snapshot at time t = 0:

History of the string at x = 8 m:

0 1 2 3 4 5

t (s)

-2

-1

0

1

2

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lta

y (

mm

)

x = 8 m0 1 2 3 4 5

t (s)

-2

-1

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1

2

Delta y

(m

m)

x = 6 m

0 2 4 6 8 10

x (m)

-2

-1

0

1

2

De

lta

y (

mm

)

t = 1 s

History at x = 6 m, for a wave travelling right at 1 m/s:

Snapshot at t=1 s of the below wave:

0 2 4 6 8 10

x (m)

-2

-1

0

1

2

Delta y

1 m/s

t = 3s

0 1 2 3 4 5

t (s)

-2

-1

0

1

2

Delta y

(m

m)

History of the string at x = 2m:

Snapshot at t=3 s of a wave travelling right at 1 m/s:

History at x = 4 m, for a wave travelling left at 1 m/s:

Snapshot at t=0 s of the below wave:

0 2 4 6 8 10

x (m)

-2

-1

0

1

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De

lta

y (

mm

)

t = 0 s

0 1 2 3 4 5

t (s)

-2

-1

0

1

2

Delta y

(m

m)

Snapshot of aLongitudinal Wave

Equlibrium position

Wave pulse

0

0 2 4 6 8 10 12

x (cm)

-1

-0.5

0

0.5

1

De

lta

x (

cm

)

Snapshot:

0

Δx is displacementfrom equlibrium

2 4 6 8 10 12

2 4 6 8 10 12

Periodic Waves

8

Up to now we’ve been playing with pulses.

Any wave with a repeating shape is a periodic wave. You can have square waves, triangular waves, sinusoidal waves, etc.

Sinusoidal waves are the most common.

• Also the most important: it turns out you can represent any wave as a combination of sinusoidal waves. (This is called Fourier analysis.)

• Sinusoidal waves are generated by moving something in simple harmonic motion.

0 1 2 3 4 5

x (m)

-1

-0.5

0

0.5

1

De

lta

y(x

, t=

0)

(mm

)

0 1 2 3 4 5

x (m)

-1

-0.5

0

0.5

1

De

lta

y(x

, t=

T/4

) (m

m)

v = 200 m/sSnapshot

at t=0

Snapshotat t=T/4

Snapshotat t=T/2

0 1 2 3 4 5

x (m)

-1

-0.5

0

0.5

1

De

lta

y(x

, t=

T/2

) (m

m)

Snapshot at t=0 with wavelength λ halved:

Snapshot at t=0 with frequency fhalved but speed unchanged:

0 1 2 3 4 5

x (m)

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, t=

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(mm

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, t=

0)

(mm

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Compression(higher density)

Rarefaction(lower density)

Compression(higher density)

Rarefaction(lower density)

Compression(higher density)

Rarefaction(lower density)

LongitudinalWaves

Periodic:

Pulse:

Equlibrium position

Wave pulse

0

0

2 4 6 8 10 12

2 4 6 8 10 12

Speaker

v

wavelength (λ)

12

What Waves AreA wave is a disturbance from equlibrium, propagating through some medium (like a string).

It takes energy to disturb a particle from equlibrium, so energy must be travelling through the medium causing the disturbance.

• Waves transport energy, not matter.

We’ll be using waves on a string as our basic model.

“Waves on a String” Simulation:http://phet.colorado.edu/simulations/sims.php?sim=Wave_on_a_String

13

Particle on a String

In a transverse sinusoidal wave, each bit of string is moving vertically in simple harmonic motion.

Particles in a fluid move back and forth in SHM as well during a longitudinal sinusoidal wave.

Fig. 16.4

Wave SpeedBy the time a particle has completed one cycle of SHM (e.g. moved down and back up), the wave has moved forward by one wavelength (λ).

The time it takes for the particle to do this is one period (T).

So, since the wave speed v is constant, v = λ/T or v = λf.

14Fig. 16.4

Example: TsunamiAn earthquake off the coast of Sumatra on 26 December, 2006, sent a tsunami smashing into southeast Asia. Satellites measured the wavelength to be about 800 km (!), and a period between waves of one hour.

• What was the wave speed, in km/h and m/s?

15

Example: UltrasoundUltrasound — sound with frequency too high for humans to hear (above about 20 000 Hz) — is a useful tool for medical imaging. Send the sound waves into the body, and listen to the echoes off of various tissues and bones etc. The speed of sound in body tissue is typically about 1500 m/s.

• To get a clear image, considering the typical size of what the doctors are looking at, the wavelength of the sound should be about 1 mm or less. What is the minimum frequency they should use?

16

Describing wave shapeWe’ve already described the way a particle in SHM moves over time (history graph!) back in the previous chapter:! y(t) = A cos(ωt + ϕ).

• Each point on the string (or whatever’s waving) is moving up and down with the same period, but with different phase. One point is at a peak, another at a trough, another moving through the equilibrium position, etc.

• In other words, each point on the string has a different initial phase ϕ, so ϕ depends on x. We can write this as “ϕ(x)”.

17

What about the shape of a wave in space at a given moment (snapshot graph!)?

It’s still sinusoidal, so we can use cosine (or sine).

In time (history graph), we had a phase that looks like “ωt + ϕ”. A phase like that gave us a nice cosine curve with the right period, and it let us choose t = 0 to be at any point in the cycle.

Let’s try something similar with position. We can’t use ω since the units are wrong (time, not space), so we have to come up with something else...

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So we need to translate a position in space (x) into a phase angle so we can use cosine.

If x = λ is one wavelength from the origin, then x/λ is the fraction of a wavelength we are away from the origin.

There are 2π radians in one cycle. So 2πx/λ should give the correct phase!

For convenience, define the wavenumber:

So in space, then: y(x) = A cos(kx + ϕ0)

Also useful: ! , so v = ω/k.

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This is not the spring constant!

v = λf =2π

k

ω

2π

Now we’d like to put it all together, to get a wavefunction that describes the wave’s behaviour in time and in space.

Each bead or bit of string is moving in simple harmonic motion, with a different phase. The phase, in other words, depends on x. So we can describe this by! y(t) = A cos(ωt + ϕ(x)).

If you move over a distance x, the phase will be different by kx = 2πx/λ. Exactly what phase you end up with depends on what phase you had at the origin, just like with time, so we need a phase constant ϕ(x=0) = ϕ0. Then ! ϕ(x) = 2πx/λ + ϕ0.

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Waves in Time and Space

Fig. 16.4But what about the sign? Should we use y(x, t) = A cos(kx – ωt) or y(x, t) = A cos(kx + ωt)? It’s not obvious!

Consider the first point that starts on axis at t=0 in the diagram to the right. Its phase at this time is 3π/2.

Increasing the phase would move this bit of string up at first. But instead, as the wave moves to the right this bit goes down!

In order to describe this, we need to use–ωt; this way the phase decreases as time increases.

So a sinusoidal wave moving to the right can be described by:

! y(x, t) = A cos(kx – ωt + ϕ0)

This is called a wavefunction.21

0 1 2 3 4 5 6

Phase (theta), radians

-1

-0.5

0

0.5

1

co

s(t

he

ta)

graph ofcos(θ)

increasing phase !

Going the Other WaySo far we’ve been describing waves moving to the right (positive x direction). What about waves moving to the left?

The behaviour of a bit of string as time proceeds is now just the opposite of what it was before. So we just need to flip the sign on the time-dependent term in our phase — i.e. replace –ωt with +ωt.

So depending on direction, our wavefunction is:

! y(x, t) = A cos(kx – ωt + ϕ0)

! y(x, t) = A cos(kx + ωt + ϕ0)

22

Sinusoidal wave moving in –x direction.

Sinusoidal wave moving in +x direction.

More on PhaseWe’ve put back the phase constant ϕ0:! y(x, t) = A cos(kx – ωt + ϕ0)

ϕ0 represents the phase at (x, t) = (0, 0).

The value (kx – ωt + ϕ0) gives the phase at any point in space and time.

• For a crest (y = +A), the phase can be 0, 2π, 4π, –2π. –4π, etc.

• For a trough (y = –A), the phase can be π, 3π, 5π, –π, –3π, etc.

v = ω/k (or λf) is often called the “phase velocity” vp.

23

Example: Birds on a StringTwo tiny birds are sitting 3.00 m apart on a long, heavy rope. Some joker comes along and starts shaking one end of the rope in SHM, with a frequency of 2.00 Hz and an amplitude of 0.075 m. The speed of the resulting wave is 12.0 m/s. At time t=0 the person’s hand is at the top of its motion (maximum positive displacement).

• What are the amplitude, angular frequency, period, wavelength, and wave number of the wave?

• What is the difference in phase between the two birds?

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So now you know:• what waves are.

• what actually travels along a wave (energy!).

• how a particle moves as a wave passes through.

• how the frequency, wavelength, and speed of a wave are related.

• how to describe a wave mathematically.

25

Aside: Partial DerivativesA partial derivative is the type of derivative you need to use when you have a function of more than one variable, such as the wavefunction y(x, t).

It works the same as a regular derivative except you replace d with ∂ (a sort of curly “d”).

It’s basically a derivative that winks at you ;-) and says, “We both know this is a function of x and t, but I’ll just pretend it’s only a function of one of them for now.” You take the derivative with respect to, say, x, treating t as just another constant (or vice versa).

26

The Wave EquationPretty much all waves follow something called the wave equation. (Don’t confuse this with the wave function, which is y(x,t) and describes a specific wave!)

The wave equation looks like this:

It’s a relationship between the curvature of the wave (at some location, at some time) and the acceleration of the particle in the wave at that same place and time.

(We’ll be using it to find v (the wave speed) for some specific types of waves.)

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(You don’t need this on your formula sheet.)

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These are the same SHM equations of motion we already know, but the initial phase (the phase when t=0) at position x is kx + ϕ0. (Compare with A cos(ωt + ϕ).)

We can also study the shape of the string (or whatever’s waving) at some specific snapshot in time, by looking at how the displacement y varies with x...

Vertical displacement of the string at x:

Vertical velocity of the string at x:

Slope of the string:

Curvature of the string:

Now this is interesting... compare to the accelleration:

Both are a constant times y. And they’re similar constants: (2π/λ)2 vs (2π/T)2.

29

Solve both the particle acceleration and the string curvature for y and equate them:

Remember that wave speed v = ω/k, so:

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The Wave Equation

The WaveEquation

Wave speed, notparticle speed!

“Applying” the Wave EquationTo get a wave equation for a particular type of wave, you need these ingredients:

• A restoring force (Fnet).

• Newton’s second law (Fnet = ma).

• Lots of linearization and simplification (calculus!).

Our purpose: once we’re done, we’ll be able to read v (the wave speed) directly from the resulting wave equation.

31

Speed of a Wave on a StringWe’ve seen that the speed of a wave on a string doesn’t depend on the amplitude or the frequency. What does it depend on?

More tension pulls the string back to equilibrium faster, so we might expect that to speed up the wave.

A heavier string will swing around more slowly, so we might expect that to slow down the wave.

• The wave doesn’t care about the total mass of the string, so we’re probably more interested in the linear mass density (mass per unit length) —"how heavy a single bit of string is.

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restoring force!

Linear Mass Density μThe linear mass density (μ) of a string is a measure of the string’s “heaviness”. Thicker ropes have a higher linear mass density than thinner ropes. Steel cables have a higher linear mass density than nylon climbing ropes. etc.

If you know the linear mass density, and you know how long your piece of rope is (L), then you know the mass of that piece of rope: m = μL.

When you cut up a rope, you change the mass of the piece you have left (less rope!), but you don’t change μ.

When you stretch a rope, you change μ but not m.

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Forces on a Bit of StringConsider a bit of string of length Δx at position x. Its mass is μΔx, and its displacement is y(x, t).

The tension in the string pulls on each end of the bit with forces of slightly different magnitudes and directions, since the string is curved.

• The horizontal components of F1 and F2 must be the same since the bit is not moving horizontally.

• The vertical components don’t quite cancel; something remains to pull the bit back to equilibrium.

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F2

F1

F1x = F

F1y

F2x = F

F2y

∆x

x x+∆x

Diagram is very exaggerated.

F2

F1

F1x = F

F1y

F2x = F

F2y

∆x

x x+∆x

To figure out the vertical components F1y and F2y, remember that the tension F1 and F2 must point along the string at that point.

So the slope of the string tells us the direction of the tension: rise/run = F1y/F or F2y/F. That is:

∂y

∂x

����x

= −F1y

F

∂y

∂x

����x+∆x

=F2y

Fand

force points down,but slope is up.

So the net vertical force on this bit of string is:

Plug this into F = ma:

Rearrange:

On the left: that’s a derivative!36

m a

�∂y∂x

���x+∆x

− ∂y∂x

���x

�

∆x=

µ

F

∂2y

∂t2

F2

F1

F1x = F

F1y

F2x = F

F2y

∆x

x x+∆x

Let’s rewrite that to make it a bit clearer.Define! (S is for Slope :)

Then for that last line:

But S(x + Δx)–S(x) is just the amount S changes when we change x. In other words, S(x + Δx) – S(x) = ΔS.

In the limit where Δx is very, very small (you’re looking at two very close bits of string) then, the definition of a derivative gives us:

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So we have a formula for the way the slope of a tiny bit of string changes as you look at different points along the wave. Let’s write that out in full.

Hey, it’s the Wave Equation! Or very similar, at least; here’s what the Wave Equation said:

So ! or

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Speed of a wave on a string.

F = tension.μ = mass per unit length ! (linear mass density).

Wave Speed vs Particle SpeedBe careful when talking about speeds in waves. There are two!

The wave is moving along at a constant speed which depends on the tension and linear density of the string.

Each particle or bit of the string is moving up and down in simple harmonic motion; its speed is changing all the time. (The formulas for the speed and acceleration of each bit of string are the same simple harmonic motion equations from before.)

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When we disturb part of a medium (such as a bit of string) from equlibrium in order to generate a wave, we’re doing work on that part of the system.

•Work is a force acting over a distance. Work transfers energy.

• By doing work on a bit of string, we put energy into the system.

When I push a part of the system out of equilibrium, that part pushes on its neighbour — doing work on its neighbour. When work is done, energy is transferred. In this way energy propagates through the system.

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Work and Energy in Waves

Power (Propagation of Work)Consider a point a on a string carrying a wave from left to right.

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, or:

Fy is negative whenslope is positive.

Slope =

y(x,t) = vertical displacement = “wavefunction”(Any other wavefunction will work, too!)

So

F1x = F

F1y

x

y

a

string

When the bit of string moves up or down, Fy does work on the bit of string to the right of point a.

• Work is a force acting over a distance. And work transfers energy.

The rate at which energy is transferred into the bit of string to the right of a is called the power (P). Since work = force x distance, we can write power = force x velocity:

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or:Rate of energy transfer for a wave on a string.

F1x = F

F1y

x

y

a

string

What we just saw made no assumptions on the shape of the wave. It holds for any wave on a string —"not just sinusoidal waves, and not even just periodic waves.

Let’s find the power in a sinusoidal wavefunction:

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Power in Sinusoidal Waves

So: Rate of energy transfer for a sinusoidal wave on a string.

We can rewrite this, using v = ω/k (or rather, k = ω/v) and v2 = F/μ (from the wave equation):

Notice that P is proportional to sin2 of something, so everything that contributes to P is positive or zero. Energy flows in the direction of the wave (which we’ve assumed to be in the +x direction here) or it doesn’t flow at all, depending on where (x) and when (t) we’re looking at the wave.

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becomes

This replaces k with properties of the string, which are usually easier to work with.

Since the only factor in our equation for power that depends on time is the sin2 function, the time-average of power depends on the average of sin2. Over a whole number of cycles, the average of sin2θ is 1/2.

Then the average power is given by:

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Average Power

Fig. 15.16(This is not a typo!)

Example: Diana, Duck of ScienceDiana, Duck of Science, is studying waves. In the name of Science! she has constructed a very, very long rope so that she can ignore reflections. She straps her feet to the free end of the rope, and flaps up and down in simple harmonic motion, at a frequency of 2.00 Hz. The amplitude of her motion is 75.0 cm.

What is the maximum (instantaneous) power Diana puts into the rope?

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Very long rope.μ = 0.250 kg/m.

Tension: F = 36.0 N.

Quark!

f = 2.00 HzA = 0.0750 m

0.250 kg/m36.0 N 2π(2.00 Hz)

0.0750 m

What is Diana’s average power?

As tenacious and stalwart a duck of Science as Diana is, she eventually tires, and her power output decreases. What is the amplitude of the wave when her average power has decreased by a factor of 100?

this stuff unchangedfrom before

Pavg is 1/100th of before

So now you know:• how to find a formula for the wave speed in a given

medium using the wave equation.

• (Don’t confuse the “wave equation” with the “wavefunction” y(x,t)!)

• what determines the speed of a wave on a string.

• how to find the rate of energy propagation (i.e. the power) carried by a wave on a string.

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Boundaries and ReflectionsWaves reflect when they hit a boundary. You’ve heard echoes, and you saw the reflections on the stretched spring in class.

• This is just the conservation of energy; the energy has nowhere to go, so it’s sent back the way it came.

How exactly they reflect depends on the boundary conditions —"the nature of the boundary. For the end of a string or spring, the two basic boundary conditions are fixed or free.

• Fixed: The string is held in place by some support.

• Free: The string is free to move up and down (but not back and forth!).

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Fixed EndReflections

When the wave reaches the fixed end of the string, it pulls on the support (whatever the string is attached to).

The support pushes back. If the displacement of that part of the wave was upward, it pushes down, and vice versa.

This has the effect of inverting the wave as it’s sent back along the string.

The wave also reverses — the front reaches the boundary first, so it will be sent back the other way first.

51 Fig. 16-18

Free EndReflections

When the wave reaches the free end of the string, the end goes up or down as the displacement of the wave dictates. (There’s nothing to prevent it from doing so.)

The end is up, meaning it’s displaced from equlibrium, so it gets pulled back by the tension or whatever the restoring force is.

This is just the normal behaviour of the wave, so the wave is sent back down the string the way it came, not inverted.

The wave is also reversed, as with the fixed end and for the same reason.

52 Fig. 16-18

Principle of SuperpositionWhen two waves overlap (exist in the same place at the same time), we simply add the displacements of each wave at each position to get the combined wave.

In other words: y(x, t) = y1(x, t) + y2(x, t).

Note that, even if y1 and y2 are sinusoidal waves, we can’t just add the amplitudes!

• In general the two waves can have different amplitudes, frequencies, wavelengths, and phase constants. The result might not even look sinusoidal.

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Principle ofSuperposition

Fig. 16-11

Standing Waves on a String

Fig. 15.23nodes

antinodes

Standing Waves result from two waves travelling in opposite directions.

Nodes are the locations where the string never leaves equlibrium.

Antinodes are the locations where the string’s amplitude is greatest.

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Wavefunction forStanding Waves

The standing wave can be described by a wavefunction just like we’ve been doing: y(x,t) describes the shape of the string’s displacement as a function of time.

The standing wave is the sum of two “travelling” waves —"the incident wave and the reflection:! y1(x,t) = A cos(kx – ωt) (incident wave moving right)! y2(x,t) = –A cos(kx + ωt) (reflected wave moving left)

The wavefunction for the standing wave is just the sum of these two wavefunctions — add the displacement of each wave at each place and time: y(x,t) = y1(x,t) + y2(x,t).

y(x,t) = y1(x,t) + y2(x,t)

! = A0 cos(kx – ωt) – A0 cos(kx + ωt)

! = A0 [cos(kx – ωt) – cos(kx + ωt)]

Trigonometric identity: ! cos(a ± b) = cos a cos b ∓ sin a sin b

So we have:y(x,t) = A0 [(coskx cosωt + sinkxsinωt) – ! ! (coskx cosωt – sinkxsinωt)]

y(x,t) = 2A0 sin(kx) sin(ωt)

or:! y(x,t) = A sin(kx) sin(ωt)

59

Wavefunction for astanding wave on a string, fixed end at x=0.

Denotes opposite signs

Note the signs!

Let’s look at that wavefunction in detail:! y(x,t) = A sin(kx) sin(ωt)

The amplitude of the standing wave is A = 2A0, or twice the amplitude of either of the original travelling waves.

A snapshot of this wave will be a sinusoidal wave with wavelength λ=2π/k, with an amplitude of A sin(ωt). In other words, the shape of the wave is always the same. (Peaks don’t move, troughs don’t move, etc.) Every part of the wave is oscillating “in phase” (at least between two nodes).

• A negative “amplitude” here (sin(ωt) can be positive or negative) just means we’ve flipped the wave over.

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We can easily determine the location (and spacing!) of nodes and antinodes from our new wavefunction:! y(x,t) = Asw sin(kx) sin(ωt)

Nodes (locations where the string doesn’t move) occur where sin(kx)=0.

• This happens when kx = 0, π, 2π, 3π, etc.

• Then nodes occur at x = 0, π/k, 2π/k, 3π/k, etc.,or x = 0, λ/2, λ, 3λ/2, etc. (k = 2π ⁄ λ)

Antinodes (locations where the string moves the most)occur where sin(kx)=1: kx = π/2, 3π/2, 5π/2, etc. This translates to x = λ/4, 3λ/4, 5λ/4, etc.

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Nodes and Antinodes Energy TransportAs an aside, there is no energy transport along a standing wave.

• Energy sloshes back and forth between antinodes and their adjacent nodes, but it doesn’t travel along the string.

• This makes sense, because a standing wave is made up of two travelling waves moving in opposite directions. Each travelling wave carries the same energy, so there’s no net energy transport.

62

Useful Case:A Clamped String

The wavefunction we’ve developed for standing waves, y(x,t) = A sin(kx) sin(ωt), fixes a node at x=0. Now let’s consider what happens if you fix the string at both ends, like you’d find in a guitar, piano, power line, etc. (I’ll refer to this as a “clamped” string. I made that up, but it’s convenient for us.)

We’ll start by looking at sinusoidal waves on a clamped string. We’ll look at more general wave shapes shortly.

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“Fixed” ⇒ NodesSince the string is held in place at both ends, the wave function has nodes at both ends, since a node is any place where the string isn’t oscillating. So if the length of the string is L, the wavefunction must have nodes at x = 0 and x = L. (There can be other nodes in between.)

Since this requirement has nothing to do with time, we look at the part of the wavefunction that describes the shape of the wave: sin(kx). We need sin(kx) = 0 at x = 0 (which we have automatically) and x = L.

• We get this if the phase (kx) at x = L to be nπ, where n is some integer.

64

So at x = L we have kL = nπ, or

Any standing wave on a clamped string must have a wavelength that satisfies this requirement.

• Other waves can exist on a clamped string, reflecting back and forth, but they won’t be standing waves.

65

kAllowed wavelengths for a standing wave on a string with nodes at x=0 and x=L.

Standing Wave WavelengthsThe set of possible wavelengths λn of the standing waves on a clamped string corresponds to a set of possible standing wave frequencies, of course, denoted fn.

where v is the speed of waves on the string.

The smallest frequency f1 corresponds to the largest possible wavelength λ1=2L: f1 = v/2L.

• This is called the fundamental frequency. Other possible frequencies are integer multiples of this: fn = nv/2L = nf1 (2v/2L, 3v/2L, etc).

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Standing Wave Frequencies

The various standing wave frequencies are also called harmonics. The fundamental frequency is also called the first harmonic or fundamental harmonic.

The series of frequencies is a harmonic series.

Harmonics above the first are often called overtones, particularly by musicians. The second harmonic is the first overtone, etc.

The string vibrates the air it’s moving in (and the instrument body it’s attached to), at the same frequency as the string’s motion. So the length of the string, and the wave speed on the string (which depends on tension and string density), determine the possible frequencies we hear.

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On a particular guitar, the speed of transverse waves on one of the strings is 143 m/s, and the distance between the bridge and the nut (i.e. the length of the vibrating part of the string) is 0.488 m.

What are the frequency and the wavelength of the fundamental harmonic?

If you replace the string with one half as heavy (i.e. half the linear mass density), what happens to f and λ?

If you gently touch the guitar string at its centre, what is the wavelength of the lowest harmonic that can still vibrate?

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Example: Guitar Harmonics

Another way of saying that a system is oscillating with a sinusoidal standing wave is that the system is oscillating in a normal mode. Each particle is moving up and down in SHM at the same frequency, in synch.

• If you have more complex waves, they’ll move back and forth across the string, so the individual bits of string move differently depending on where the wave is — and they won’t necessarily be in SHM.

Each standing wave frequency is a normal mode.

These get a special name because they’re the “natural frequencies” of the system, and respond to resonance.

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Normal Modes A String in Slow Motion

Video from University of Salford (UK).http://www.acoustics.salford.ac.uk/feschools/waves/guitarvideo.htm

Stringed InstrumentsFor strings, we’ve seen how the wave speed v relates to the tension F and the linear density μ:

Plug this into the formula for the nth harmonic fn to see how the string affects the frequency of the normal modes. For the fundamental (first) harmonic:

So you can get a lower note by using a longer string, a heavier string, or a looser string.

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Fundamental frequencyfor a string fixed atboth ends.

The strings in a piano vary in both length and thickness. The lowest strings are actually wrapped with wire to lower their frequency.

Bigger instruments make lower notes.

The lower-note strings on a guitar are made differently than the higher notes. (On my guitar the low strings are steel wound with bronze, and the high strings are nylon.) This is particularly important because all the strings are the same length.

Strings are tuned by adjusting their tensions.

Wind instruments work basically the same way. But that’s a story for the next chapter...

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• what happens to a wave when it reflects at a boundary.

• how to combine waves when they overlap.

• what causes a “standing” wave, and how to describe one mathematically.

• how clamping a string at both ends restricts its possible standing waves to very specific wavelengths and frequencies.

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So now you know: