Given:
The wave equation is:
y(x,t)=3.0sin(36t+0.018x+π4)y(x,t) = 3.0 \sin \left( 36t + 0.018x + \frac{\pi}{4} \right)
where:
• y(x,t)y(x,t) is the displacement in cm,
• xx is in cm,
• tt is in seconds.
Now, let's answer the questions step by step.
Part A: Is this a travelling wave or a stationary wave? If it is a travelling wave, what is the speed and direction of its propagation?
The general form of a travelling wave is:
y(x,t)=Asin(kx−ωt+ϕ)y(x,t) = A \sin(kx - \omega t + \phi)
or
y(x,t)=Asin(kx+ωt+ϕ)y(x,t) = A \sin(kx + \omega t + \phi)
where:
• AA is the amplitude,
• kk is the wave number,
• ω\omega is the angular frequency,
• ϕ\phi is the phase constant.
Comparing the given equation with the general form, we can identify the following:
y(x,t)=3.0sin(36t+0.018x+π4)y(x,t) = 3.0 \sin \left( 36t + 0.018x + \frac{\pi}{4} \right)
We can see that the wave has a term involving both xx and tt, which indicates this is a travelling wave (not a stationary wave, which would have only sine terms of xx and tt that are separated).
To find the speed of the wave, we use the relation:
v=ωkv = \frac{\omega}{k}
From the wave equation:
• ω=36 rad/s\omega = 36 \, \text{rad/s},
• k=0.018 rad/cmk = 0.018 \, \text{rad/cm}.
Now, calculate the speed vv:
v=ωk=360.018=2000 cm/sv = \frac{\omega}{k} = \frac{36}{0.018} = 2000 \, \text{cm/s}
The wave is moving in the negative x-direction because of the +0.018x+ 0.018x term in the phase. This indicates the wave is traveling to the left.
So the wave is a travelling wave with a speed of 2000 cm/s and moving in the negative x-direction.
Part B: What is its amplitude and frequency?
The amplitude AA is the coefficient of the sine function, which is 3.0 cm.
The angular frequency ω=36 rad/s\omega = 36 \, \text{rad/s}.
To find the frequency ff, we use the relation:
f=ω2πf = \frac{\omega}{2\pi}
Substitute ω=36\omega = 36:
f=362π≈5.73 Hzf = \frac{36}{2\pi} \approx 5.73 \, \text{Hz}
Thus, the amplitude is 3.0 cm, and the frequency is approximately 5.73 Hz.
Part C: What is the initial phase at the origin?
To find the initial phase at x=0x = 0, substitute x=0x = 0 into the wave equation:
y(0,t)=3.0sin(36t+π4)y(0,t) = 3.0 \sin(36t + \frac{\pi}{4})
The initial phase at x=0x = 0 is simply the phase term in the sine function when x=0x = 0, which is:
Phase=π4\text{Phase} = \frac{\pi}{4}
Thus, the initial phase at the origin is π4\frac{\pi}{4} radians.
Part D: What is the least distance between two successive crests in the wave?
The distance between two successive crests is the wavelength λ\lambda, which can be calculated using the formula:
λ=2πk\lambda = \frac{2\pi}{k}
Substitute k=0.018 rad/cmk = 0.018 \, \text{rad/cm}:
λ=2π0.018≈349.1 cm\lambda = \frac{2\pi}{0.018} \approx 349.1 \, \text{cm}
Thus, the least distance between two successive crests is approximately 349.1 cm.
Summary of Answers:
1. Type of wave: It is a travelling wave with a speed of 2000 cm/s in the negative x-direction.
2. Amplitude: 3.0 cm
3. Frequency: Approximately 5.73 Hz
4. Initial phase at the origin: π4\frac{\pi}{4} radians
5. Least distance between two successive crests (wavelength): Approximately 349.1 cm
