Transverse and Longitudinal Waves — Question 8

Question 8

An ideal fluid cannot sustain a static shear restoring force

For this question, an ideal fluid has no shear rigidity but does resist compression. A bulk disturbance would require either a restoring force from transverse shear or one from compression.

1. Which bulk mechanical-wave type can this model support, and why?

2. Does the existence of ripples on a free fluid surface contradict the bulk conclusion? Identify a different possible restoring mechanism.

Question 8 - Solution

An ideal fluid cannot sustain a static shear restoring force

Method and mathematical working

Ask which deformation produces a restoring force under the model assumptions. Compression changes pressure; ideal bulk shear does not create a restoring shear stress.$$\begin{aligned}G=0&\Rightarrow\text{no static shear restoring stress},\\K>0&\Rightarrow\delta p=-K\frac{\delta V}{V}\quad\text{for small compression}.\end{aligned}$$

(a) Calculation and conclusion

The model can support longitudinal compressional disturbances: changing density produces pressure forces. It cannot support ordinary bulk elastic shear waves because the assumed shear restoring force is absent.

(b) Interpretation and limits

No. A free surface adds boundary geometry; gravity and surface tension can restore its deformation. Surface-wave particle motion need not be purely longitudinal. The conclusion about bulk shear waves under the ideal-fluid assumptions does not exclude surface waves.

Physical check

The free surface introduces gravity and surface tension as additional force agents. Their presence changes the physical problem rather than contradicting the bulk-fluid conclusion.