Wavelength and Frequency — Question 3

Question 3

A fixed interface preserves repetition rate but changes spacing

A continuous wave crosses a stationary interface from a region with speed $6\,\mathrm{m/s}$ into one with speed $4\,\mathrm{m/s}$. Its incident wavelength is $.30\,\mathrm{m}$. Assume a transmitted wave at the same periodic interface motion.

1. Find the transmitted frequency and wavelength.

2. Explain why retaining the same wavelength would be inconsistent with the interface motion.

Question 3 - Solution

A fixed interface preserves repetition rate but changes spacing

Method and mathematical working

The displacement at a stationary junction has one time dependence. Enforce this common repetition rate before using either region's speed.$$\begin{aligned}f_2&=f_1=\frac{v_1}{\lambda_1},\\ \frac{\lambda_2}{\lambda_1}&=\frac{v_2}{v_1}.\\[4pt]f_1=f_2&=6/0.30=20\,\mathrm{Hz},\qquad \lambda_2=4/20=0.20\,\mathrm m.\end{aligned}$$

(a) Calculation and conclusion

The incident frequency is $6/.30=20\,\mathrm{Hz}$. A stationary interface is driven through twenty cycles each second, so the transmitted wave also has $f=20\,\mathrm{Hz}$. Its wavelength is $4/20=.20\,\mathrm{m}$.

(b) Interpretation and limits

If the transmitted wavelength remained $.30\,\mathrm{m}$, its frequency would be $13.33\,\mathrm{Hz}$, inconsistent with the common periodic displacement at the boundary. The changed travel distance per cycle accounts for the wavelength change.

Physical check

The slower region must have a smaller wavelength if the frequency is unchanged. This qualitative ordering should agree with the numerical result.