A sinusoidal wave traveling in the negative x direction (to the left) has an amplitude of 20.0 cm, a wavelength
of 35.0 cm, and a frequency of 12.0 Hz. The transverse position of an element of the medium at t = 0, x = 0 is y
= - 3.00 cm, and the element has a positive velocity here. We wish to find an expression for the wave function
describing this wave.
To find an expression for the wave function, we can use the general form of a sinusoidal wave:
y(x, t) = A * sin(kx - ωt + φ)
where:
- A is the amplitude of the wave
- k is the wave number, given by k = 2π/λ (λ is the wavelength)
- ω is the angular frequency, given by ω = 2πf (f is the frequency)
- φ is the phase constant
Given information:
- A = 20.0 cm (amplitude)
- λ = 35.0 cm (wavelength)
- f = 12.0 Hz (frequency)
- t = 0, x = 0, y = -3.0 cm (transverse position at t = 0, x = 0)
To determine the phase constant (φ), we can substitute the given values of x, t, and y into the wave equation.
At t = 0, x = 0, y = -3.0 cm:
-3.0 = 20.0 * sin(0 - 0 + φ)
-3.0 = 20.0 * sin(φ)
Now, we can solve for the value of φ:
sin(φ) = -3.0 / 20.0
φ = arcsin(-3.0 / 20.0)
Using a calculator, the value of φ is approximately -0.1504 radians.
Now, we can substitute the values of A, k, ω, and φ into the wave equation to find the expression for the wave function:
y(x, t) = 20.0 * sin((2π/35.0) * x - (2π * 12.0 * t) - 0.1504)
Therefore, the expression for the wave function describing this wave is:
y(x, t) = 20.0 * sin((2π/35.0) * x - (24πt) - 0.1504)