sound wave of frequency 166 Hz travels with a speed 332 ms−1
along positive
x-axis in air. Each point of the medium moves up and down through 5.0 mm.
Write down the equation of the wave and calculate the (i) wavelength, and (ii)
wave number. How far are points which differ in phase by 45°? (2+3)
ii) Derive expression for average rate at which energy is transported by a progressive
wave propagating in a medium.
To write down the equation of the wave, we need to know the general form of a wave equation:
y = A sin(kx - ωt + φ)
Where:
- y represents the displacement of the point in the medium at a given position x and time t.
- A is the amplitude of the wave, which is 5.0 mm in this case.
- k is the wave number, which we will calculate later.
- ω is the angular frequency, calculated as 2πf, where f is the frequency in Hz.
- φ is the phase constant.
(i) To calculate the wavelength, we can use the formula:
wavelength (λ) = speed (v) / frequency (f)
In this case, the speed of sound in air is given as 332 ms^(-1), and the frequency is 166 Hz.
Therefore, the wavelength is:
λ = 332 ms^(-1) / 166 Hz = 2.0 m
(ii) The wave number (k) can be calculated using the formula:
k = 2π / λ
Substituting the value of λ we just calculated:
k = 2π / 2.0 m = π m^(-1)
Now, let's move on to the distance between points that differ in phase by 45°.
When two points on a wave differ in phase by 45°, it means that the phase difference (Δφ) between them is 45° or π/4 radians.
To calculate the distance (Δx) between these points, we can use the formula:
Δx = Δφ / k
Substituting Δφ = π/4 and k = π m^(-1):
Δx = (π/4) / π m^(-1) = 0.25 m
Therefore, points that differ in phase by 45° are 0.25 meters apart.
Now, let's move on to deriving the expression for the average rate at which energy is transported by a progressive wave:
The average power (P) carried by a progressive wave is given by:
P = (1/2)ρAvω^2
Where:
- ρ is the density of the medium through which the wave is propagating.
- A is the amplitude of the wave.
- ω is the angular frequency of the wave.
The energy is related to power by the equation:
Energy (E) = Power (P) × Time (t)
To find the average rate at which energy is transported, we divide the energy by time:
Average rate of energy transport = E / t
Substituting the expression for power into the energy equation:
Average rate of energy transport = (1/2)ρAvω^2 × t / t
Simplifying this expression:
Average rate of energy transport = (1/2)ρAvω^2
This is the derived expression for the average rate at which energy is transported by a progressive wave propagating in a medium.