Sound and doppler effect on window pressure

[lesismore100]

Homework Statement

*See attached diagram*

On straight, level, parallel tracks separated by a distance d,
two trains are testing their horns (in still air of density ρair).
The horns (located at the train fronts) emit equal frequencies.
Horn 1 is a pipe, open at one end, emitting a total power P
and resonating at its 9th harmonic. Horn 2 is a loudspeaker
of circular diameter equal to the length of horn 1.
In one test, the trains (1 and 2) and three researchers (A, B, C) are all stationary and are positioned as shown.
Sound from horn 1 takes time t to reach C, midway between the tracks.
A hears only horn 1 (loudness = β1). B (right next to A) hears both. But when both horns are sounding from
the positions shown, C hears both horns at maximum combined loudness. And if train 2 were repositioned
farther and farther forward along its track until it was exactly side-by-side with train 1, C would also hear
maximum combined loudness at 18 other positions of train 2 (including the fi nal position when the trains were
exactly side-by-side).
In a second test, C stands alone, still midway between the tracks. The trains (from much farther away) move
toward her at constant speeds (v2 > v1, but only v1 is known), both sounding their horns. When the two trains
are side-by-side, C notes a beat frequency of f beat .
Find the net air force (magnitude & direction) on a window pane (area = A2) on the right side of train 2.
Train 2ʼs windows were closed just before it started moving.

The list of known values: d, ρ(air) (density of air) , P (power), t, β1 , v1 , f beat , A2

Homework Equations

sin(theta)=1.22(wavelength)/Diameter
F=AxP
fc=f2[1/(1-v2/v)]

The Attempt at a Solution

By working backward I know I have to find the velocity of train 2 in order to find the F exerted by the wind on the window. In order to find the velocity I need to use the doppler shift equation, however in order to find the frequency of train 2 I need the wavelength of its train horn.

Any help would be appreciated. Thanks.

 

[anathema]

Hello,

Thank you for your post. Based on the information given, I believe the following steps may help you find a solution to this problem:

1. Start by drawing a diagram of the situation, including the positions of the trains, researchers, and the window pane.

2. Use the given information to calculate the wavelength of the sound emitted by horn 1. This can be done using the equation sin(theta)=1.22(wavelength)/Diameter, where theta is the angle at which the sound is heard by researcher C (midway between the tracks).

3. Use the given information to calculate the frequency of horn 1. This can be done by dividing the speed of sound (which can be found using the density of air and the given formula for the speed of sound) by the calculated wavelength.

4. Since horn 1 resonates at its 9th harmonic, you can use this information to find the fundamental frequency of the horn. This can be done by dividing the calculated frequency by 9.

5. Use the given information to calculate the speed of train 1. This can be done by dividing the distance between the two trains by the time it takes for the sound to travel from horn 1 to researcher C.

6. Now that you have the speed of train 1, you can use the given beat frequency to find the speed of train 2. This can be done using the equation fc=f2[1/(1-v2/v)], where fc is the beat frequency, f2 is the frequency of horn 2, and v is the speed of sound.

7. Use the calculated speed of train 2 to find the wavelength of the sound emitted by horn 2. This can be done using the equation sin(theta)=1.22(wavelength)/Diameter, where theta is the angle at which the sound is heard by researcher C (midway between the tracks).

8. Use the calculated wavelength of horn 2 to find its frequency. This can be done by dividing the speed of sound by the calculated wavelength.

9. Now that you have the frequencies of both horns, you can use the given information to find the combined loudness at the various positions of train 2. This can be done by using the formula for combining sound levels (βcombined = 10log(10^(β1/10)+10^(β2/10))).

10. Finally, use the calculated frequency of horn 2 to find the wavelength