Types of Waves — Question 8

Question 8

A wave can propagate on a surface rather than through a volume

A short disturbance spreads from a point along an ideal long string in one experiment and outward across a broad water surface in another. Ignore boundaries during the observation.

1. Compare the dimensions of the regions through which these disturbances propagate.

2. Why does circular spreading on the water not imply that each marked surface particle travels outward with the expanding circle?

Question 8 - Solution

A wave can propagate on a surface rather than through a volume

Method and mathematical working

Count the independent spatial directions available to the advancing disturbance. This classification concerns the region of propagation, not the number of components of local particle motion.$$\begin{aligned}\text{string: }q&=q(x,t),\qquad \text{surface: }q=q(x,y,t),\\r_{\mathrm{front}}(t)&=vt\quad\text{does not require }r_{\mathrm{particle}}(t)=vt.\end{aligned}$$

(a) Calculation and conclusion

The string disturbance propagates along a one-dimensional path, in two possible directions. The surface disturbance spreads over a two-dimensional region, with a circular front in an isotropic uniform setting. This classifies the propagation domain, not the vector direction of local motion.

(b) Interpretation and limits

The circle marks the advancing disturbance, while local water-particle motions must be measured separately. A particle can move around a small local path as the front passes. Actual water waves can also have drift, so the pattern alone establishes neither zero drift nor outward transport at the front speed.

Physical check

A circular front can move outward while material particles follow local trajectories near their original positions. Tracking a marked particle is a separate experiment from photographing the front.