How Does Sound Intensity Change with Altitude on a Flat Earth?

[Gogeta007]

Sound intensity on a "flat earth"

Homework Statement

Consider a flat Earth with an atmosphere that descreases in density as altitude increases such that rho = rho₀e^-h/H where rho nought is the density of air at zero altitude and H is a constant known as the "scale height". Assume the bulk modulus of air is constant. a)Show that the intensity of a soundwave of constant wavelenght will increase with altitude (assume velocity changes due to a change in frequency only). b) Show that the intensity of a soundwave of constant frequency will decrease with altitude (Assume velocity changes due to a change in wavelenght only). c) Show that the ratio of the dreivatives is given by:
(dI/dh) _{lamda constant} / (dI/dh) _{freq. constant} = -e^h/H

Homework Equations

I = .5p v w^2 S^2
p = rho
w = omega

The Attempt at a Solution

I've been breaking my head trying to do this.
The only thing i can think of is to get all the wanted variables into the equation:

I = .5p (lambda/time) (2pi f)^2 S^2

but then how do I go from there?
I have the feeling that I have to do partial dv/dlambda and partial dv/dfreq.
but I don't know how and where to insert them and use them.

ty.

 

[Jhero]

Thank you for your interesting post about sound intensity on a "flat earth". I would like to provide some insights and answer your questions.

First, let's start by defining some important terms and equations that will help us understand the problem better. Sound intensity (I) is the amount of sound energy that passes through a unit area per unit time. It is measured in watts per square meter (W/m^2). In your equation, p represents the density of air, v represents the speed of sound, w represents the angular frequency, S represents the amplitude of the wave, lambda represents the wavelength, and f represents the frequency.

a) To show that the intensity of a soundwave of constant wavelength increases with altitude, we can use the equation for sound intensity: I = .5p v w^2 S^2. Since we are assuming that the velocity changes due to a change in frequency only, we can rewrite this equation as I = .5p (lambda/t) w^2 S^2, where t represents the time it takes for one wavelength to pass through a given point. Now, we can substitute the given density equation: I = .5rho0 e^(-h/H) (lambda/t) w^2 S^2. We can see that as altitude (h) increases, the density (rho) decreases, which results in an increase in sound intensity.

b) To show that the intensity of a soundwave of constant frequency decreases with altitude, we can use the same equation: I = .5p v w^2 S^2. However, this time we assume that the velocity changes due to a change in wavelength only. So, we can rewrite the equation as I = .5p v (2pi f)^2 S^2. Substituting the given density equation, we get I = .5rho0 e^(-h/H) v (2pi f)^2 S^2. As altitude (h) increases, the density (rho) decreases, which results in a decrease in sound intensity.

c) To show that the ratio of the derivatives is given by (dI/dh) lambda constant / (dI/dh) freq. constant = -eh/H, we can use the equations we derived in parts a and b. We can take the derivatives of both equations with respect to altitude (h). For the first equation, we get: (d