(b) Sketch the graph of displacement against distance graph for progressive wave above ina range of 0 ≤ 𝑥 ≤ 𝜆 at time, t = 0 s.

To sketch the graph of displacement against distance for a progressive wave over the range $0 \leq x \leq \lambda$ at time $t = 0$ seconds, follow these steps:

1. ### Break Down the Problem

• Understand that a progressive wave can be represented by a sine or cosine function.
• For $t = 0$, we can use the general form of the wave equation $y(x, t) = A \sin(kx)$ or $y(x, t) = A \cos(kx)$, where:
  • $A$ is the amplitude,
  • $k$ is the wave number, defined as $k = \frac{2\pi}{\lambda}$,
  • $\lambda$ is the wavelength.

2. ### Relevant Concepts

• The sine function varies between -1 and 1; therefore, the displacement $y$ will vary between $-A$ and $A$.
• The graph will represent one complete cycle of the wave from $0$ to $\lambda$.

3. ### Analysis and Detail

• For $t = 0$, the wave described by $y(x) = A \sin(kx)$ will give: $y(x) = A \sin\left(\frac{2\pi}{\lambda} x\right)$
• At $x = 0$: $y(0) = A \sin(0) = 0$
• At $x = \frac{\lambda}{4}$: $y\left(\frac{\lambda}{4}\right) = A \sin\left(\frac{2\pi}{\lambda} \cdot \frac{\lambda}{4}\right) = A \sin\left(\frac{\pi}{2}\right) = A$
• At $x = \frac{\lambda}{2}$: $y\left(\frac{\lambda}{2}\right) = A \sin(\pi) = 0$
• At $x = \frac{3\lambda}{4}$: $y\left(\frac{3\lambda}{4}\right) = A \sin\left(\frac{3\pi}{2}\right) = -A$
• At $x = \lambda$: $y(\lambda) = A \sin(2\pi) = 0$

4. ### Verify and Summarize

• The graph will start from $0$ at $x = 0$, reach the maximum amplitude $A$ at $x = \frac{\lambda}{4}$, return to $0$ at $x = \frac{\lambda}{2}$, drop to $-A$ at $x = \frac{3\lambda}{4}$, and return to $0$ at $x = \lambda$, depicting one complete oscillation.

Final Answer

The graph of displacement against distance for the progressive wave at $t = 0$ is a sine wave that oscillates between $-A$ and $A$ over the interval $0$ to $\lambda$. The key points are:

• At $x = 0$, $y = 0$
• At $x = \frac{\lambda}{4}$, $y = A$
• At $x = \frac{\lambda}{2}$, $y = 0$
• At $x = \frac{3\lambda}{4}$, $y = -A$
• At $x = \lambda$, $y = 0$