Question

two speakers are separated by d = 7.0 m (consider speakers are on y - axis) and are being driven by a signal generator at an unknown frequency. a student with a sound level meter walks on (starting from one speaker) x - axis x = 5.0 m and then y - axis y = 3.0 m and finds the first minimum intensity. wave speed of the sound is v = 343.0 m/s.
a) what is the frequency supplied by the signal generator?
b) if frequency decreases by 20%, what is the new changing path length.

Explanation:

Step1: Determine the path - length difference for the first minimum

For destructive interference at the first minimum, the path - length difference $\Delta r=r_2 - r_1=\frac{\lambda}{2}$.
Let the two speakers be at $y_1 = 0$ and $y_2 = 7.0$ m on the y - axis. A student is at $x = 5.0$ m and $y = 3.0$ m.
The distance from the first speaker $r_1=\sqrt{x^{2}+(y - 0)^{2}}=\sqrt{5^{2}+3^{2}}=\sqrt{25 + 9}=\sqrt{34}\approx5.83$ m.
The distance from the second speaker $r_2=\sqrt{x^{2}+(y - 7)^{2}}=\sqrt{5^{2}+(3 - 7)^{2}}=\sqrt{25+16}=\sqrt{41}\approx6.40$ m.
The path - length difference $\Delta r=r_2 - r_1=\sqrt{41}-\sqrt{34}\approx6.40 - 5.83 = 0.57$ m. Since $\Delta r=\frac{\lambda}{2}$, then $\lambda = 2\Delta r=1.14$ m.

Step2: Calculate the frequency

We know that the speed of sound $v = 343$ m/s and the formula $v = f\lambda$. So, $f=\frac{v}{\lambda}$. Substituting $v = 343$ m/s and $\lambda = 1.14$ m, we get $f=\frac{343}{1.14}\approx300.88$ Hz.

Step3: Analyze the new frequency and wavelength

If the frequency decreases by 20%, the new frequency $f'=(1 - 0.20)f = 0.8f$. Since $v = f\lambda=f'\lambda'$ and $v$ is constant, $\lambda'=\frac{f}{f'}\lambda=\frac{1}{0.8}\lambda = 1.25\lambda$.
The new path - length difference for the first minimum is still $\Delta r'=\frac{\lambda'}{2}=1.25\times\frac{\lambda}{2}$. Since the position of the minimum is related to the path - length difference, and the geometry of the setup remains the same in terms of the relative positions of the speakers and the observer, the new position of the minimum will change. But if we just consider the new path - length difference for the first minimum in terms of the new wavelength, it is related to the original path - length difference by a factor of 1.25.

Answer:

a) The frequency supplied by the signal generator is approximately $301$ Hz.
b) The new path - length difference for the first minimum is 1.25 times the original path - length difference.