Wave Motion

(a) 33 m

In one vibration the sound travels one wavelength: $\lambda = \dfrac{v}{f} = \dfrac{352}{384} = 0.917$ m. In 36 vibrations it travels $36\lambda = 36 \times 0.917 \approx 33$ m.

(D) $\lambda$

By definition, the wavelength is the distance between two nearest particles that are in the same phase of vibration (for example, two consecutive crests).

Stationary waves: When two progressive waves of the same amplitude, frequency and speed travel along the same line in opposite directions, they superpose to form a wave pattern that does not move forward. This is a stationary (standing) wave. Some points (nodes) are always at rest and some points (antinodes) vibrate with maximum amplitude.

Proof that node-to-node distance is $\lambda/2$:

1. Let the incident and reflected waves be

$$y_1 = a\sin(\omega t - kx), \qquad y_2 = a\sin(\omega t + kx), \qquad k = \frac{2\pi}{\lambda}$$

2. By the principle of superposition,

$$y = y_1 + y_2 = a[\sin(\omega t - kx) + \sin(\omega t + kx)]$$

3. Using $\sin C + \sin D = 2\sin\dfrac{C+D}{2}\cos\dfrac{C-D}{2}$:

$$y = 2a\cos kx \,\sin\omega t = A\sin\omega t, \qquad A = 2a\cos\frac{2\pi x}{\lambda}$$

So each particle does SHM with amplitude $A$ that depends on its position $x$.

4. Nodes are points where $A = 0$:

$$\cos\frac{2\pi x}{\lambda} = 0 \;\Rightarrow\; \frac{2\pi x}{\lambda} = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \dots$$ $$x = \frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, \dots$$

5. Distance between two consecutive nodes:

$$\frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{\lambda}{2}$$

Hence, the distance between two consecutive nodes is $\boldsymbol{\lambda/2}$. (In the same way, antinodes at $x = 0, \lambda/2, \lambda, \dots$ are also $\lambda/2$ apart.)

(C) Diagram (C)

When the pulses fully overlap, the displacements add (principle of superposition). Over the left unit: $+2 + (-1) = +1$; over the right unit: $+1 + (-1) = 0$. So the result is an upward pulse of height 1 unit over the left half and a flat string over the right half, which is diagram (C).

 Units:         left unit   right unit
 Pulse 1:          +2          +1
 Pulse 2:          -1          -1
 Sum:              +1           0

 Result:     ___
 ___________|   |___________   (height 1 over left unit only)