The average pressure of a wave is related to its properties, including density, volume, amplitude, and angular frequency. The mathematical proof of the relationship can be derived using the principles of wave mechanics.
Consider a one-dimensional progressive wave traveling in a medium, such as a sound wave. The pressure variation at a specific point in the wave can be described by a sinusoidal function:
P(x, t) = P0 * sin(kx - ωt)
Where: P(x, t) is the pressure at position x and time t. P0 is the amplitude of the pressure wave. k is the wave number, related to the wavelength λ as k = 2π / λ. x is the spatial coordinate. ω is the angular frequency, related to the frequency f as ω = 2πf. t is the time.
To find the average pressure over one period of the wave, we integrate the pressure function over one complete cycle, from t = 0 to t = T, where T is the period of the wave.
1/T ∫[0 to T] P(x, t) dt = 1/T ∫[0 to T] P0 * sin(kx - ωt) dt
Since sin(kx - ωt) is a periodic function with a period of 2π/ω, the integration can be simplified by changing the limits of integration accordingly:
1/T ∫[0 to T] P(x, t) dt = 1/(2π/ω) ∫[0 to 2π/ω] P0 * sin(kx - ωt) dt
Applying the trigonometric identity for the average of sine function over one cycle:
1/(2π/ω) ∫[0 to 2π/ω] sin(kx - ωt) dt = P0/2
Thus, we find that the average pressure of the wave over one period is equal to half of the amplitude (P0) of the wave.
Now, let's consider the relationship between the wave properties and the average pressure. The amplitude of the pressure wave (P0) is related to the amplitude of the displacement of the wave particles (A). For a sound wave, the amplitude of particle displacement is directly proportional to the pressure amplitude.
The density of the medium (ρ) and the volume of the medium (V) can be combined to give the mass (m) of the medium, which is constant:
m = ρV
The angular frequency (ω) of the wave is related to the wave speed (v) and the wave number (k) as ω = vk. Using the wave speed equation v = λf = ω/k, we can express ω as ω = v*k.
Substituting the above expressions into the average pressure equation, we have:
Average Pressure = (1/2) * P0 = (1/2) * (A * ρV * ω^2) = (1/2) * (A * ρV * (v*k)^2) = (1/2) * (A * ρV * v^2 * k^2) = (1/2) * (A^2 * ρ * V * k^2 * v^2)
Since the product (ρ * V) represents the mass per unit volume (ρ) times volume (V), it can be replaced by the letter "m":
Average Pressure = (1/2) * (A^2 * m * k^2 * v^2)
Thus, we arrive at the final expression for the average pressure, which is equal to half of the density (ρ), volume (V), square of the amplitude (A), and square of the angular frequency (ω).