Question

7. The first resonant length of a closed air column occurs when the length is 40 cm. What will the third resonant length be? 

8. A police car with a siren that has a frequency of 900 Hz is moving toward you at 32.0 m/s. Given that the speed of sound is 340 m/s, what is the apparent frequency you hear?

Answer 1

7) There is the following length-wavelength relationship for the first harmonic of the closed air column:

$$\lambda_1=4L_1=4\cdot0.4\ m=1.6\ m.$$

Finally, we can find the third resonant length:

$$L_3=\dfrac{5}{4}\lambda_1=\dfrac{5}{4}\cdot1.6\ m=2\ m.$$

8) We can find the apparent frequency that the observer hear from the Doppler Shift formula:

$$f_o=f_s(\dfrac{v}{v-v_s}),$$ $$f_o=900\ Hz\cdot(\dfrac{340\ \dfrac{m}{s}}{340\ \dfrac{m}{s}-32\ \dfrac{m}{s}})=994\ Hz.$$