Question

A wave travels along a stretched horizontal rope. The vertical distance from crest to trough for this wave is 13 cm and horizontal distance from crest to trough is 28 cm.
(A) What is the wavelength of this wave?

(B) What is the amplitude of this wave?

Answer 1

Answer:

(A) The wavelength of this wave is $56\; \rm cm$.

(B) The amplitude of this wave is $6.5\; \rm cm$.

Explanation:

Refer to the diagram attached. A point on this wave is at a crest or a trough if its distance from the equilibrium position is at a maximum.

The amplitude of a wave is the maximum displacement of each point from the equilibrium position. That's the same as the vertical distance between the crest (or the trough) and the equilibrium position.

• On the diagram, the distance between the two gray dashed lines is the vertical distance between a crest and a trough. According to the question, that distance is $\rm 13\; \rm cm$ for the wave in this rope.
• On the other hand, the distance between either gray dashed line and the black dashed line is the distance between a crest (or a trough) and the equilibrium position. That's the amplitude of this wave.

Therefore, the amplitude of the wave is exactly $\displaystyle \frac{1}{2}$ the vertical distance between a crest and a trough. Hence, for the wave in this question,

$\begin{aligned}& \text{Amplitude}\\ &= \frac{1}{2} \times (\text{Vertical distance between crest and trough}) \\ &= \frac{1}{2} \times 13\;\rm cm = 6.5\; \rm cm\end{aligned}$.

The wavelength of a transverse wave is the same as the minimum (horizontal) distance between two crests or two troughs. That's twice the horizontal distance between a crest and a trough in the same period.

$\begin{aligned}& \text{Wavelength}\\ &= 2 \times (\text{Horizontal distance between adjacent crest and trough}) \\ &= 2 \times 28\;\rm cm = 56\; \rm cm\end{aligned}$.