3 Problems involving superposition

mst3kjunkie

Here are the three problems I'm having trouble with:

1. The problem statement, all variables and given/known data
Two identical loudspeakers separated by distance d emit 170 Hz sound waves along the x-axis. As you walk along the axis, away from the speakers, you don't hear anything even though both speakers are on. What are three possible values for d? Assume a sound speed of 340 m/s.

2. Relevant equations

phase difference = 2*Pi*(change in r/wavelength)+initial phase difference=2(m+1/2)Pi

interference is destructive if the path-length difference r=(m+1/2)wavelength

3. The attempt at a solution

find the wavelength

f=v/2L = v/wavelength
wavelength=v/f =340/170 =2

d=(m+1/2)wavelength

for m=0: d=1m
m=1: d=2m
m=2 d=3m

did I do this correctly?

1. The problem statement, all variables and given/known data
A steel wire is used to stretch a spring. An oscillating magnetic field drives the steel wire back and forth. A standing wave with three antinodes is created when the spring is stretched 8.0cm. What stretch of the spring produces a standing wave with two antinodes

2. Relevant equations

unsure

3. The attempt at a solution
the wave is in the third harmonic. The wavelength can be found using

wavelength=4L/m =4(.08)/3 = 8/75

I'm not sure how to get to the book's answer of 18 cm.

1. The problem statement, all variables and given/known data
Two loudspeakers 5.0m apart are playing the same frequency. If you stand 12.0 m in from of the plane of the speakers, centered between them, you hear a sound of maximum intensity. As you walk parallel to the plane of the speakers, staying 12.0m in front of them, you first hear a minimum of sound intensity when you are directly in from of one of the speakers.

a. what is the frequency of the sound? Assume a sound speed of 340 m/s
b. If you stay 12.0m in front of one of the speakers, for what other frequencies between 100 Hz and 1000 Hz is there a minimum sound intensity at this point?

2. Relevant equations
unsure for part b

3. The attempt at a solution
I've solved part a, f=170 Hz.

I don't even know how to start part b.

Chi Meson

First problem:

Quote by mst3kjunkie

3. The attempt at a solution

find the wavelength

f=v/2L = v/wavelength
wavelength=v/f =340/170 =2

d=(m+1/2)wavelength

for m=0: d=1m
m=1: d=2m
m=2 d=3m

did I do this correctly?

Excessive busywork led you to the correct final equation, then you did the simple math wrong. plug in your m's again.

Chi Meson

2nd problem

Quote by mst3kjunkie

the wave is in the third harmonic. The wavelength can be found using

wavelength=4L/m =4(.08)/3 = 8/75

I'm not sure how to get to the book's answer of 18 cm.

You are looking for the stretch of the spring. Goto Hookes' Law. How does a change in tension affect the fundimental frequncy of a standing wave in a string?

prace

Isn't the [tex]\lambda[/tex] for the third harmonic [tex]\frac{2L}{3}[/tex]?

If you use hookes law, don't we need to know the spring constant k before using this information to solve this problem?

Chi Meson

You are right about the 2L/3, although this thread "died" 9 months ago.

But to answer your questions, you don't need to know the spring constant. You just need to know the proportionality between the stretch of the string, to the tension of the spring, to the speed of the wave in the string.

If the frequency is constant, by doubling the stretch of the spring, the tension will double, and the wave speed will increase by 1.41 (square root 2).

to have the same frequency go from being the 3rd harmonic to being the 2nd harmonic (2 is 2/3 of 3), the speed will have to increase by 3/2 of it's original, and the tension and therefore stretch will have to increase by the square root of 3/2

Hmm, but that doesn't give 18 cm.

Just a minute...d'oh!

the stretch will increase by the square of 3/2 (not the root).

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prace	3 Problems involving superposition Isn't the [tex]\lambda[/tex] for the third harmonic [tex]\frac{2L}{3}[/tex]? If you use hookes law, don't we need to know the spring constant k before using this information to solve this problem?