The wave equation describes the mathematical relationship between the various properties of a wave. In the case of a plane progressive wave, the wave equation can be expressed as:
y(x, t) = A * sin(kx - ωt + φ)
where:
- y(x, t) represents the displacement of the wave at position x and time t.
- A is the amplitude of the wave (given as 3m in your question).
- k is the wave number, which is related to the wavelength λ by the equation: k = 2π / λ.
- x represents the position along the direction of propagation.
- ω is the angular frequency of the wave, given by ω = 2πf, where f is the frequency (given as 150Hz in your question).
- t represents time.
- φ is the phase constant, which determines the initial phase of the wave.
In order to determine the value of k, we need to know the wavelength of the wave. The wavelength can be calculated using the wave velocity (v) and the frequency (f) as follows:
v = λf
Rearranging the equation, we get:
λ = v / f
Substituting the given values, we have:
λ = 220m/s / 150Hz ≈ 1.47m
Now we can find the value of k:
k = 2π / λ = 2π / 1.47m ≈ 4.28 rad/m
Finally, the wave equation for the given plane progressive wave with an amplitude of 3m, a frequency of 150Hz, and a velocity of 220m/s is:
y(x, t) = 3 * sin(4.28x - 942πt + φ)