Beats and Doppler

Mahi Tuatahi

Glue in and complete the beats explanation cloze activity.

Recall: Doppler Shift

• A change in wave frequency due to the motion of emitter or observer
• Moving towards gives a higher frequency, and away a lower frequency

Beats Caused by Doppler

• Because movement of a sound source can cause a change of frequency (Doppler effect), this can cause beats to occur.
• For example, two identical devices where one is moving towards/away from the other can cause a slight change in frequency, causing beats to arise.

Pātai Toru

Both Josh and Zak are holding identical devices emitting a frequency of \(100Hz\). Josh sprints away from him again while Zak listens. Zak observes a beat frequency of \(3Hz\). The speed of sound is \(343ms^{-1}\).

SHOW that Josh is walking at a speed of \(v_{s} = 10.6ms^{-1}\).

Step 1: Calculate \(f'\) that Zak is observing from Josh’s device (\(f_{b} = |f_{2} - f_{1}|\))

Step 2: Use this value to find \(v_{s}\) in \(f' = f\frac{v_{w}}{v_{w} \pm v_{s}}\)

Whakatika Toru a)

\[ \begin{aligned} f_{b} &= |f_{2} - f_{1}| \newline f &= 103 \text{ or } 97 \end{aligned} \]

Because Josh is sprinting away, the frequency should be lower, so \(97Hz\).

Whakatika Toru b)

\[ \begin{aligned} f' &= f \frac{v_{w}}{v_{w} \pm v_{s}} \text{(+ because away)} \newline 97 &= 100 \frac{343}{343 + v_{s}} \newline \frac{97}{100} &= \frac{343}{343 + v_{s}} \newline (343 + v_{s}) \times 0.97 &= 343 \newline 343 + v_{s} &= \frac{343}{0.97} \newline v_{s} &= \frac{343}{0.97} - 343 = 10.6ms^{-1} \end{aligned} \]

Practice

1. Homework Booklet Doppler Q8c