Sound Waves — Question 6

Question 6

A two-medium travel time determines the layer thickness

A pulse traverses a straight $1.00\,\mathrm m$ path normal to two fluid layers. Layer A has speed $1500\,\mathrm{m/s}$, layer B speed $1000\,\mathrm{m/s}$. The measured one-way time is $0.800\,\mathrm{ms}$. Ignore interface delays.

1. Infer the length occupied by each layer.

2. What travel-time range is physically possible for this fixed total path? Use it to assess a claimed $0.60\,\mathrm{ms}$ result.

Question 6 - Solution

A two-medium travel time determines the layer thickness

(a) Method and calculation

Let $x$ be the length in A. Times add:$$t=\frac{x}{1500}+\frac{1-x}{1000}.$$$$x=\frac{0.001-0.000800}{1/1000-1/1500}=0.600\,\mathrm m.$$The length in B is $0.400\,\mathrm m$; the two contributions are $0.400\,\mathrm{ms}$ each.

(b) Interpretation and further calculation

The constraints $0\le x\le1$ impose$$\frac1{1500}\le t\le\frac1{1000},\quad0.667\le t\le1.000\,\mathrm{ms}.$$A $0.60\,\mathrm{ms}$ time falls below even the all-fast-medium limit. It is inconsistent with the stated path and speeds, so averaging the two speeds cannot repair it.

Physical check

The effective speed is total distance divided by total time, $1250\,\mathrm{m/s}$, between the component speeds.