The Speed of Sound — Question 7

Question 7

A speed gradient requires integrating slowness

Along a straight path $0\le x\le L$, local sound speed is $c(x)=c_0(1+\alpha x/L)$ with $c_0=300\,\mathrm{m/s}$, $L=600\,\mathrm m$ and $\alpha=0.20$. Assume local propagation and no ray bending.

1. Integrate the travel time and calculate the effective path speed.

2. Compare with using the arithmetic mean of the endpoint speeds. Show that the expression has the correct uniform-medium limit.

Question 7 - Solution

A speed gradient requires integrating slowness

(a) Method and calculation

Each short segment contributes $dt=dx/c(x)$:$$t=\int_0^L\frac{dx}{c_0(1+\alpha x/L)}=\frac{L}{c_0\alpha}\ln(1+\alpha).$$$$t=10\ln1.2=1.8232\,\mathrm s,\qquad c_{\mathrm{eff}}=L/t=329.09\,\mathrm{m/s}.$$

(b) Interpretation and further calculation

The endpoint arithmetic mean is $(300+360)/2=330\,\mathrm{m/s}$, giving $t=1.8182\,\mathrm s$ and underestimating the delay. Expanding,$$\frac{\ln(1+\alpha)}{\alpha}=1-\frac\alpha2+\frac{\alpha^2}3+\cdots,\quad t\to L/c_0.$$Travel time averages reciprocal speed along the path, so slow regions receive extra weight.

Physical check

The effective speed lies between $300$ and $360\,\mathrm{m/s}$. The formula requires $1+\alpha x/L>0$ throughout.