Path Difference Question

Gordon has decided to replace his bat signal with a sound system, after receiving complaints from Batman that a light in the sky lacked visibility. In order for the sound system to maximize in effectiveness, Gordon has installed two sound systems at a distance of 6km apart and at the edge of the city and has them pointed to the East, where the batcave apparently lies.

A] Gordon, after discussing this sound system with Batman, has been given this hand-drawn map of the batcave’s location. Knowing the information written in this map, what is the path difference of the two sound systems?

B] Based on this path difference, how will Batman hear the sound compared to 1 sound system?

C] Batman, having been completely annoyed by the obnoxious sound systems, has decided to relocate his layer to a point where he won’t need to hear it any longer. However, due to limited open space to relocate and a lack of funds, Batman only wants to move his batcave along the y-axis and the minimum distance possible. What would be the closest point (x,y) where Batman can fulfill the above conditions and live in peace?

Path Difference Question

Gordon has decided to replace his bat signal with a sound system, after receiving complaints from Batman that a light in the sky lacked visibility. In order for the sound system to maximize in effectiveness, Gordon has installed two sound systems at a distance of 6km apart and at the edge of the city and has them pointed to the East, where the batcave apparently lies.

A] Gordon, after discussing this sound system with Batman, has been given this hand-drawn map of the batcave’s location. Knowing the information written in this map, what is the path difference of the two sound systems?

After careful inspection of this quality map, we can plot the two sound systems along with the batcave on a graph depicting the distance the batcave is away from both sound systems (courtesy of Batman):

We can then calculate the distance using the formula:

   d 2− d1 =  √x  y  ) 2+ ( + 3 2−√x  (y  3) 2+   −   2

   d 2− d1 =  √(12)  (0)  ) 2+ ( + 3 2−√(12)  ((0)  3) 2+   −   2

 d 2− d1 =  √144+9−√144 9+  

. Therefore, the path difference is 0. d 2− d1 = 0

B] Based on this path difference, how will Batman hear the sound compared to 1 sound system?

The path difference between 2 waves can be related to the integer multiple of the wavelength, with complete constructive interference being represented by:

  λd 2− d1 = n

And destructive interference being represented by:

+ (½)  λd 2− d1 = n

In this case, the path difference found above was 0. Therefore, since 0 is an integer multiple of , theλn batcave is positioned at point where complete constructive interference occurs, resulting in the sound being twice as loud as one sound system.

C] Batman, having been completely annoyed by the obnoxious sound systems, has decided to relocate his layer to a point where he won’t need to hear it any longer. However, due to limited open space to relocate and a lack of funds, Batman only wants to move his batcave along the y-axis and the minimum distance possible. What would be the closest point (x,y) where Batman can fulfill the above conditions and live in peace?

From what has already been noted above, we know that the path difference must fulfill the equation:

+ (½)  λd 2− d1 = n

Since we want the minimum amount of movement possible, we will use the value of 0.5km as our .λn

.5   √x  y  ) 2+ ( + 3 2−√x  (y  3) 2+   −   2 = 0

.5   √12  y  ) 2+ ( + 3 2−√12  (y  3) 2+   −   2 = 0

Using a graphing calculator, we find that:

y = +/-1.034

Which means that Batman needs to move his batcave by either 1.034km North or South in order to experience destructive interference and end his suffering.