Waves and Sound — Hard Practice Quiz

A Physics cheat sheet for Waves and Sound — every key formula with its symbols defined — plus a hard-level practice quiz to test recall.

Practice quiz

1. A string of length $L$ and mass $m$ is under tension $F$. It vibrates at its fundamental frequency $f$. If the tension is quadrupled to $4F$ and the length is halved to $L/2$, while the mass $m$ remains constant, what is the new fundamental frequency $f'$ in terms of $f$?

   • $\frac{f}{\sqrt{2}}$
   • $2f$
   • $2\sqrt{2}f$
   • $4f$

   Answer: $2\sqrt{2}f$

2. A spherical sound source emits power $P$. At a distance $r$ from the source, the sound intensity level is $\beta$. If the distance from the source is doubled to $2r$, by how much does the sound intensity level change?

   • It decreases by $3 \text{ dB}$
   • It decreases by $6 \text{ dB}$
   • It decreases by $10 \text{ dB}$
   • It decreases by $20 \text{ dB}$

   Answer: It decreases by $6 \text{ dB}$

3. A wave on a string has a speed $v$. If the tension $F$ is kept constant, but the string's material is replaced with one that has twice the mass for the same length, how must the wavelength $\lambda$ change to maintain the original frequency $f$?

   • $\lambda$ must be halved.
   • $\lambda$ must be multiplied by $\sqrt{2}$.
   • $\lambda$ must be divided by $\sqrt{2}$.
   • $\lambda$ must be doubled.

   Answer: $\lambda$ must be divided by $\sqrt{2}$.

4. A sound source emits power $P$. At a distance $r_1$, the sound intensity level is $\beta_1$. If the intensity level drops by $20 \text{ dB}$ at a distance $r_2$, what is the ratio $r_2/r_1$?

   • $2$
   • $4$
   • $10$
   • $100$

   Answer: $10$

5. A stationary observer hears a frequency $f_o$ from a sound source moving towards them at speed $v_s$. If the source were instead moving away from the observer at the same speed $v_s$, and the speed of sound in the medium doubled to $2v$, what would be the new observed frequency $f_o'$ in terms of $f_o$? Assume $v_s = v/4$.

   • $\frac{1}{2} f_o$
   • $\frac{2}{3} f_o$
   • $\frac{3}{4} f_o$
   • $\frac{4}{5} f_o$

   Answer: $\frac{2}{3} f_o$

6. A string of length $L$ and mass $m$ is under tension $F$. If the string is replaced by another string of the same material (same linear density $\mu$) but twice the length ($2L$) and under four times the tension ($4F$), how does the wavelength of the fundamental mode change if the frequency is kept constant?

   • The wavelength remains the same.
   • The wavelength is halved.
   • The wavelength is doubled.
   • The wavelength is quadrupled.

   Answer: The wavelength is doubled.

7. Two identical spherical sound sources, each emitting power $P$, are placed $2r$ apart. What is the sound intensity level at the midpoint between them, relative to the intensity level from a single source at distance $r$? Assume no interference.

   • It increases by $3 \text{ dB}$.
   • It increases by $6 \text{ dB}$.
   • It increases by $10 \text{ dB}$.
   • It increases by $20 \text{ dB}$.

   Answer: It increases by $3 \text{ dB}$.

8. An observer is moving towards a stationary sound source. The observed frequency is $f_o$. If the observer's speed is $v_o$, and the speed of sound is $v$, derive an expression for the source's actual frequency $f_s$ in terms of $f_o$, $v_o$, and $v$.

   • $f_s = f_o (1 + \frac{v_o}{v})$
   • $f_s = f_o (\frac{v}{v - v_o})$
   • $f_s = f_o (\frac{v}{v + v_o})$
   • $f_s = f_o (\frac{v + v_o}{v})$

   Answer: $f_s = f_o (\frac{v}{v + v_o})$

9. A string of length $L$ and mass $m$ is under tension $F$. If the string is replaced by one of the same length but made of a material with half the linear density, and the tension is adjusted such that the wave speed remains the same, how does the new tension $F'$ compare to the original tension $F$?

   • $F' = 2F$
   • $F' = \frac{F}{\sqrt{2}}$
   • $F' = \frac{F}{2}$
   • $F' = F$

   Answer: $F' = \frac{F}{2}$

10. A sound source emits $100 \text{ W}$ of power uniformly in all directions. What is the distance from the source where the sound intensity level is $80 \text{ dB}$? Use $I_0 = 1.00 \times 10^{-12} \text{ W/m}^2$.

    • $\frac{100}{\sqrt{\pi}} \text{ m}$
    • $\frac{1000}{2\sqrt{\pi}} \text{ m}$
    • $\frac{1000}{\pi} \text{ m}$
    • $\frac{100}{2\pi} \text{ m}$

    Answer: $\frac{1000}{2\sqrt{\pi}} \text{ m}$