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 x = 0.0 cm, t = 2.0 s, we can use the equation for the displacement of a harmonic wave:
y(x, t) = A * sin(kx - ωt)
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, given by k = 2π/λ, where λ is the wavelength.
x is the position along the wave.
ω is the angular frequency, given by ω = 2πf, where f is the frequency of the wave.
t is the time.
Given values:
A = 4.4 cm
v (velocity) = 41.0 cm/s
λ (wavelength) = 45.0 cm
x = 0.0 cm
t = 2.0 s
To calculate the wave number, k:
k = 2π/λ = 2π/45.0 cm ≈ 0.139 cm^(-1)
To calculate the angular frequency, ω:
v = λf
f = v / λ ≈ 41.0 cm/s / 45.0 cm ≈ 0.911 Hz
ω = 2πf ≈ 2π * 0.911 Hz ≈ 5.72 rad/s
Now we can calculate the displacement at x = 0.0 cm, t = 2.0 s:
y(0.0, 2.0) = 4.4 cm * sin(0.139 cm^(-1) * 0.0 cm - 5.72 rad/s * 2.0 s)
y(0.0, 2.0) = 4.4 cm * sin(-5.72 rad/s * 2.0 s)
y(0.0, 2.0) = 4.4 cm * sin(-11.44 rad)
y(0.0, 2.0) ≈ 4.4 cm * sin(-11.44 rad)
y(0.0, 2.0) ≈ 4.4 cm * (-0.194) ≈ -0.85 cm
Therefore, the displacement due to the wave at x = 0.0 cm, t = 2.0 s is approximately -0.85 cm.
To calculate the displacement at x = 10.0 cm, t = 20.0 s:
Using the same values for A, v, λ, and ω from the previous calculations, we can plug them into the equation to find the displacement:
y(10.0, 20.0) = 4.4 cm * sin(0.139 cm^(-1) * 10.0 cm - 5.72 rad/s * 20.0 s)
y(10.0, 20.0) = 4.4 cm * sin(1.39 cm - 114.4 rad)
y(10.0, 20.0) ≈ 4.4 cm * sin(1.39 cm - 1790.1 rad)
y(10.0, 20.0) ≈ 4.4 cm * sin(-1788.71 rad)
y(10.0, 20.0) ≈ 4.4 cm * (0.195) ≈ 0.86 cm
Therefore, the displacement due to the wave at x = 10.0 cm, t = 20.0 s is approximately 0.86 cm.
To calculate the displacement of the wave at a specific point and time, we can use the equation for a harmonic wave:
y(x, t) = A * sin(kx - ωt + φ)
where:
- A is the amplitude of the wave
- x is the position along the x-axis
- t is the time
- k is the wave number (2π/λ, where λ is the wavelength)
- ω is the angular frequency (2πf, where f is the frequency)
- φ is the phase constant
In this case, we are given that:
- A = 4.4 cm (amplitude)
- v = 41.0 cm/s (speed)
- λ = 45.0 cm (wavelength)
- x = 0.0 cm (position)
- t = 2.0 s (time)
To calculate the wave number k, we can use the formula:
k = 2π/λ
k = 2π/45.0 cm ≈ 0.139 cm^(-1)
To calculate the angular frequency ω, we can use the formula:
ω = 2πf
Since we know the speed v and the wavelength λ, we can calculate the frequency f using the formula:
v = fλ
f = v/λ = 41.0 cm/s / 45.0 cm ≈ 0.911 Hz
ω = 2π(0.911 Hz) ≈ 5.719 rad/s
Now we have all the values needed to calculate the displacement at x = 0.0 cm and t = 2.0 s in the equation:
y(0.0 cm, 2.0 s) = 4.4 cm * sin(0.139 cm^(-1) * 0.0 cm - 5.719 rad/s * 2.0 s + φ)
To solve for the phase constant φ, we are given that the displacement is zero at x = 0 and t = 0. Therefore, substituting these values into the equation, we get:
0 = 4.4 cm * sin(φ)
sin(φ) = 0
This means that the phase constant φ can be any multiple of π since sin(φ) = 0 when φ = nπ, where n is an integer.
For x = 0.0 cm and t = 2.0 s, we can use the equation:
y(0.0 cm, 2.0 s) = 4.4 cm * sin(0.139 cm^(-1) * 0.0 cm - 5.719 rad/s * 2.0 s + φ)
Since sin(φ) = 0, the displacement will be 0 cm.
So, the displacement of the wave at x = 0.0 cm, t = 2.0 s is 0 cm.
To calculate the displacement at x = 10.0 cm and t = 20.0 s, we can use the same equation:
y(10.0 cm, 20.0 s) = 4.4 cm * sin(0.139 cm^(-1) * 10.0 cm - 5.719 rad/s * 20.0 s + φ)
Again, we need to find the phase constant φ. Since we know that sin(φ) = 0, the displacement at x = 10.0 cm, t = 20.0 s will also be 0 cm.