Sound Interference - Radio Broadcast

[dvolpe]

Homework Statement

Roger in ship offshore listening to baseball game on his radio. He notices there is destructive interference when seaplanes from nearby Coast Guard station fly directly overhead at elevations of 777 m, 970 m, and 1163 m. The broadcast station is 98 km away. Assume there is a 180 degree phase shift when the EM waves reflect from the seaplanes. What is the frequency of the broadcast?

Homework Equations

dsin theta = (m + 1/2)*lambda
d = 193 (distance between each elevation)
frequency = speed of sound/wavelength

The Attempt at a Solution

I am assuming that theta = 0 since the planes fly directly over Roger's ship. So sin theta = 1. I want to solve for lambda but don't understand how m fits in with the different elevations. Help?

 

[SammyS]

The interference radio wave (electro-magnetic wave) interference, not sound wave interference.

 

[eczeno]

it is not clear what angle you are calling theta. anyway, it is not too important. try to picture this. two signals are beamed from the radio station, one travels in a straight line to the boat, the other heads slightly upward, hits the plane, and then heads straight down to the boat. you can maybe see how these paths form a right triangle. now you just have to compare the lengths of those paths (don't forget the phase shift!) for the different altitudes and find a frequency that will give destructive interference for all those path lengths (don't forget the phase shift!).

hope that helps

 

[dvolpe]

Thank you eczeno! I found the lengths of the paths using pythagorean theorem but am unsure what to do with them next. Are they the wavelengths?

 

[eczeno]

these are not the wavelengths, in fact there will be only one wavelength, which will be that of the radio signal. this will be related to the frequency of the signal by lambda*f=c, where c is the speed of light.

your job is to figure out the frequency, which will be easy if we know the wavelength, so that is what we will focus on.

destructive interference occurs when the two signals are 180 degrees out of phase. say the path lengths of the 2 signals are d1 and d2. if d1 = d2 + m*lambda, then the signals are in phase and we get constructive interference. but if d1 = d2 + (m+1/2)*lambda, then we get destructive interference.

there is a slight complication in your problem since one signal is reflected off of a surface, and therefore undergoes a 180 degree phase shift during its travels. this is accounted for by simply dropping the +1/2 from the above formula.

 

[dvolpe]

Thank you again..I am guessing that di is the sum of the path of the signal to the plane plus the elevation back down to Roger and that d2 is the path from the station straight to the ship. What do I do with m since it is unknown? I am concerned about negative signs here and am assuming I use the destructive interference formula above but take out the 1/2, so it really looks like the constructive interference formula. Is that correct?

Why does the signal undergo a 180 degree phase shift when it is reflected off of the plane? I would just like to understand this better.

 

[eczeno]

the choice for d1 and d2 is unimportant (m could very well be a negative integer, that is ok).

you are right that with the 180 degree shift, the formulas just switch (this is only true for 180 degree shift, not say a 90 degree shift!).as to why the phase shift happens, well there are many levels of depth with which that question could be answered. at the deepest level (so far), there is the quantum electrodynamics explanation. That is beyond my abilities (i am just beginning to start to barley understand it). a simper problem which is somewhat analogous is that of a mechanical wave (in a string, say) reflecting from a fixed point. Here, all you need, really, is Newton's third law. picture a pulse traveling down a string, an upward pointing pulse. as this pulse approaches the fixed point, the string is trying to pull the fixed point up. well, this means the fixed point will pull the string down, and so the pulse will be 'pulled under' by the fixed point. now picture this happening to a wave, each crest becomes a trough, and each trough a crest, which is exactly the same effect a 180 degree phase shift has, so it is a 180 degree phase shift.

of course a light wave interacting with a surface is much more complicated, but you can imagine a similar dance between electromagnetic fields and electrons all sort of 'pulling and pushing' each other (at least that is what I imagine).

 

[dvolpe]

Ok..but what do I do with the equation..I can plug in values for d1 and d2 but am still left with m unknown as well as frequency...I am getting confused. Thanks for that explanation on the reasoning.

 

[eczeno]

d1 is fixed (lets call the straight line distance d1) and you have three d2's. so, there will be three 'path length differences', if that makes sense. let's call them Da, Db, and Dc. each of these must be an integer multiple of lambda. with the condition that those integers (these are the m's) must be consecutive (not explicitly stated, but seems to be implied in the problem), only one lambda will do the trick.

 

[eczeno]

ok, so for there to be destructive interference, $d_2-d_1=m\lambda$ for some integer m. we have three values for $d_2$, let's call them $d_a$, $d_b$ and $d_c$. these should correspond to three consecutive values of m, so

$$d_a-d_1=m\lambda$$

$$d_b-d_1=(m+1)\lambda$$

$$d_c-d_1=(m+2)\lambda$$

you can find both m and $\lambda$ with these equations.