The wave equation describes the relationship between the physical properties of a wave, such as its amplitude, frequency, velocity, and position. The equation for a plane progressive wave can be expressed as:
y(x, t) = A * sin(kx - ωt + φ)
In this equation:
- y represents the displacement of the wave at a given position (x) and time (t).
- A is the amplitude of the wave, which is given as 3m in your case.
- k is the wave number, related to the wavelength (λ) by the equation k = 2π/λ.
- x represents the position along the wave.
- ω is the angular frequency, related to the frequency (f) by the equation ω = 2πf.
- t represents time.
- φ is the phase constant, which determines the starting phase of the wave.
To find the values of k and ω, we can use the relationship between velocity, frequency, and wavelength: v = f * λ
Given:
- Frequency (f) = 150 Hz
- Velocity (v) = 220 m/s
We can rearrange the equation to solve for the wavelength (λ): λ = v / f
Substituting the values: λ = 220 m/s / 150 Hz λ ≈ 1.467 m
Using the value of wavelength, we can determine the wave number: k = 2π / λ k ≈ 4.28 rad/m
Now, we have the values of A, k, ω, and φ to plug into the wave equation:
y(x, t) = 3 * sin(4.28x - (2π * 150)t + φ)
Please note that the phase constant (φ) was not provided in the question, so the equation above includes it as a variable. The phase constant determines the initial phase of the wave and can be adjusted accordingly based on the given conditions.