A harmonic wave moving in the positive x direction has an amplitude of 4.4 cm, a speed of 41.0 cm/s, and a wavelength of 45.0 cm. Assume that the displacement is zero at x = 0 and t = 0.
Calculate the displacement (in cm) due to the wave at x = 0.0 cm, t = 2.0 s.
Calculate the displacement due to the wave at x = 10.0 cm, t = 20.0 s.
To calculate the displacement due to the wave at a specific point and time, we need to use the equation for a harmonic wave:
y(x, t) = A * sin(kx - ωt + φ)
Where:
y(x, t) is the displacement at position x and time t,
A is the amplitude of the wave,
k is the wave number (2π/λ, where λ is the wavelength),
x is the position of the point,
ω is the angular frequency (2πf, where f is the frequency),
t is the time, and
φ is the phase constant.
First, let's find the values of k and ω:
Given:
Amplitude (A) = 4.4 cm
Wavelength (λ) = 45.0 cm
Speed (v) = 41.0 cm/s
The wave number (k) can be calculated using the formula k = 2π/λ:
k = 2π/45.0 = 0.13962634 cm^-1
The angular frequency (ω) can be calculated using the formula ω = 2πf, and the frequency (f) can be found using the formula f = v/λ:
f = v/λ = 41.0/45.0 ≈ 0.9111 Hz
ω = 2πf ≈ 5.7273 radians/s
The equation of the harmonic wave becomes:
y(x, t) = 4.4 * sin(0.13962634x - 5.7273t + φ)
To find the phase constant (φ), we need to know the initial conditions. The problem states that the displacement is zero at x = 0 and t = 0. This means the phase constant is zero, φ = 0.
Using the given information, we can now calculate the displacement for the two given points:
1. At x = 0.0 cm, t = 2.0 s:
y(0, 2) = 4.4 * sin(0.13962634 * 0 - 5.7273 * 2 + 0)
y(0, 2) = -4.4 * sin(11.4546)
y(0, 2) ≈ -3.6984 cm
Therefore, the displacement is approximately -3.6984 cm at x = 0.0 cm, t = 2.0 s.
2. At x = 10.0 cm, t = 20.0 s:
y(10, 20) = 4.4 * sin(0.13962634 * 10 - 5.7273 * 20 + 0)
y(10, 20) = 4.4 * sin(-106.3994)
y(10, 20) ≈ -4.3511 cm
Therefore, the displacement is approximately -4.3511 cm at x = 10.0 cm, t = 20.0 s.