Transverse Waves: Analyzing the Displacement Function y(x, t) = f(x − at)
Question:
For a transverse wave on a string, the displacement of the string is described by the equation:
y(x, t) = f(x − at)
where f is a given function and a is a positive constant. Which of the following statements does NOT necessarily follow from this wave function?
- A. The shape of the string at time t = 0 is given by f(x).
- B. The shape of the waveform does not change as it moves along the string.
- C. The waveform moves in the positive x-direction.
- D. The speed of the waveform is a.
- E. The speed of the waveform is x/t.
Answer:
✅ A. True:
When t = 0, the displacement becomes y(x, 0) = f(x). So the shape of the string is exactly described by f(x) at that instant.
When t = 0, the displacement becomes y(x, 0) = f(x). So the shape of the string is exactly described by f(x) at that instant.
✅ B. True:
Since the displacement is entirely a function of (x − at), the waveform moves without altering its shape. This is a characteristic of a traveling wave.
Since the displacement is entirely a function of (x − at), the waveform moves without altering its shape. This is a characteristic of a traveling wave.
✅ C. True:
The term x − at shows the wave moves in the positive x-direction. If it were x + at, it would move in the negative x-direction.
The term x − at shows the wave moves in the positive x-direction. If it were x + at, it would move in the negative x-direction.
✅ D. True:
To find wave speed, consider a point of constant phase: let x − at = constant. Differentiating both sides with respect to time t gives:
To find wave speed, consider a point of constant phase: let x − at = constant. Differentiating both sides with respect to time t gives:
d(x − at)/dt = 0 ⇒ dx/dt = a
This confirms that the wave moves with speed a.
❌ E. False:
The expression x/t is not a reliable or general way to determine wave speed. The actual wave speed comes from phase velocity analysis and is specifically defined as a in this context.
The expression x/t is not a reliable or general way to determine wave speed. The actual wave speed comes from phase velocity analysis and is specifically defined as a in this context.
Conclusion:
The statement that does NOT necessarily follow from the wave function y(x, t) = f(x − at) is:
“The speed of the waveform is x/t.”
“The speed of the waveform is x/t.”