Chapter 16: Waves-(I)

General introduction to the topic of waves:

Do you have an idea how the scorpion finds its prey?

What is a wave?

Note: presentation main figures are that of Halliday and Resnick 6th edition unless otherwise specified.

Topics in Chapter 16

• Waves and Particles (How does energy travel from one point (A) to another point (B)?

• Types of Waves (transverse and longitudinal)• Sinusoidal waves description and important

relations (frequency, wavelength, wave speed and tension etc)

• Energy & Power Transfer by wave• Superposition (Interference) of waves• Standing waves and Resonance

16.1 Waves and Particles:

How does energy travel from one point (A) to another point (B)?

Particles carry energy from one point to another. You studied (in physics 101 that a ball that is thrown in projectile motion, for example, has kinetic energy and potential energy, and that one can increase at the expense of the other…etc.

17.1 Waves and Particles: (cont’d)

Waves (the disturbances) can also carry energy from one point to another while the ‘particles’’ move/ vibrate/ oscillate only locally.

The air molecules, in the case of a sound wave, for example, do not move from my mouth to you ears. But the disturbance does!!

16.2 Types of Waves:

We (in H&R) classify waves into three main categories. Can anyone tell a type of waves?

1- Mechanical waves. 2- Electromagnetic waves. 3- Matter waves.

In phys-102, we will concentrate on:

Mechanical Waves

Are there differences between these types of waves?

16.3 Transverse and Longitudinal Waves:

In a transverse wave: The direction of the wave (disturbance) is perpendicular to the direction of (local) motion of the medium.

Let’s apply this to the example of a wave moving on a taught rope.

16.3 Transverse and Longitudinal Waves: (cont’d)

A slinky can demonstrate both: transverse and longitudinal waves.

Surface ocean waves is a mixture of both.

In a longitudinal wave: The direction of the wave (disturbance) is in the same the direction of (local) motion of the medium.

Let’s apply this to the example of a sound wave moving in an air column.

Now let’s explain how a scorpion finds its prey!

vt = 50, vl = 150, t = 4.0 ms

16.4 Wavelength and Frequency:

It would be nice to ‘describe’ the waveform (shape of the wave) through some mathematical function.

In a string moving forward, the displacement (y) up-down is described through a function: y = h (x, t).

16.4 Wavelength and Frequency: (cont’d)

If the disturbance is sinusoidal, then:Y (x, t) = Ym sin (k x – t)

Let’s understand the concepts of amplitude, wavelength, [angular] wave number, period, frequency, angular frequency, and phase.

waves

Important relations

Sinusoidal Wave repeats itself after each and sine function also repeats itself after 2

Sin (kx1 + 2 ) = Sin k(x1 + ) for t = o 2 = k k = 2 / called angular wave number= radians/m

Similarly wave repeats itself for each time period T so

Sin (t1 + 2 ) = sin (t1 + T) for x= 0 T = 2 or = 2 / T but T = I/f Hence f = 2 /

16.5 The Speed of a Traveling Wave:

Let’s see how fast the wave travels; e.g., how fast the crest moves. As the points on the crest of wave retains their position (y displacement) during the movement of wave so phase which gives that displacement is constant (do not confuse with points on string as they move ) k x– t = constant

v = /k

v = /T

v = f

The wave moves one wavelength per period!!

The speed of the wave (v) is dx / dt; therefore,

k dx/dt =

Transverse velocity

Y (x, t) = Ym sin (k x – w t)

• u = dy/dt = - Ym cos (k x – w t)

This is velocity of element of string in the y direction and do not confuse this with wave velocity which is constant velocity at which wave form travels along x-axis.

• Prove that a= du/dt = - 2 Y= -2 Ym sin (k x – w t)

16.6 Wave Speed on a Stretched String:

Use dimensional analysis and/ or see the proof on page 379

A string with linear mass density , under a tension has a speed:

v

What happens to the speed when the frequency increases?

Example:

For the wave on a string in the figure below [taken at time t = 0],

1. Determine the amplitude, wavelength, angular wave number and phase angle (for the sin wave).

2. You are told that the string has a LMD(µ) of 40 g/cm and is under 10 N of tension, find the period and angular frequency of the wave.

-1 -0.5 0.5 1 1.5 2xm

-10

-5

5

10

ym

Mathematica output

16.7 Energy and Power of a Traveling String Wave:

Where is the kinetic energy minimum/maximum?

Where is the elastic potential energy minimum/maximum?

[see the proof on page 381-2]

Pavg = ½ v 2 y2m

16.7 Energy and Power of a Traveling String Wave: (cont’d)

In exams, we play games with the students.

For example, what happens if we increase the tension on the string by a factor of 9?

Pavg = ½ v 2 y2m

The average power transmitted in the wave depends on the linear mass density, on the speed, on the square of the frequency and on the square of the amplitude.

Interaction question:

What happens if we increase the wavelength by a factor of 10, keeping the tension constant?

16.9 The Principle of Superposition of Waves:

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

Two (or more) overlapping waves algebraically add to produce a resultant (or, net) wave. The overlapping waves do not alter the motion of each other.

Let’s see this superposition Mathematica code

Yres(x,t) = Ym sin (kx-t) +Ym(sin kx- t +)

As Sin +sin = 2 sin ½ ( + )cos1/2( - )

Yres(x,t) = 2Ym { cos 1/2 } (sin kx- t +)

For = 180, Ym (x,t) = 0

16.10 Interference of Waves:

Two waves propagating along the same direction with the same amplitude, wavelength and frequency, but differing in phase angle will interfere with each other in a nice way.

Let’s see the (same) waves Mathematica code

Checkpoint #5:

16.12 Standing Waves:

What happens when two waves propagating in opposite directions with the same amplitude, wavelength and frequency will interfere with each other such as to create standing waves!!

Let’s see the (same) waves Mathematica code

16.12 Standing Waves: (cont’d)

Reflection at a Boundary:

1- Hard Reflection

2- Soft Reflection

You’re going to love Mathematica; see this code.

Reflection-Transmission

Checkpoint #6:

16.13 Standing Waves and Resonance:When a string of length L is clamped between two points, and sinusoidal waves are sent along the string, there will be many reflections off the clamped ends.

At specific frequencies, interference will produce nodes and large anti-nodes.

We say we are at resonance, and that the string is resonating at resonant frequencies.

Let’s see the (same) waves

16.13 Standing Waves and Resonance: (cont’d)

Example: A 75 cm long string has a wave speed of 10 m/s, and is vibrating in its third harmonic. Find the distance between two adjacent anti-nodes.

The fundamental mode (n=1) has a fundamental frequency: f1 = v/(2L)

The second harmonic (n=2) has a frequency: f2 = 2 f1 = v/(L)

The nth harmonic has a frequency: fn = n f1 = nv/(2L)

The wavelength of the nth harmonic is: n = 2L/n

What is the distance between two adjacent nodes (or anti-nodes)?

What is the distance between a node and its neighboring anti-node?