IMO Practice Test — Waves
12 Questions • 15 min • Olympiad level
15:00
Question 1 of 12
A string under tension $T$ has wave speed $v$. To double $v$, the tension must be made:
2 times
4 times
$\sqrt{2}$ times
half
Explanation: $v\propto\sqrt{T}$, so doubling $v$ needs $T$ to be quadrupled.
Question 2 of 12
A closed pipe and an open pipe have the same fundamental frequency. The ratio of their lengths (closed:open) is:
1:1
1:2
2:1
1:4
Explanation: Closed $f_1=\frac{v}{4L_c}$, open $f_1=\frac{v}{2L_o}$; equal $\Rightarrow\frac{v}{4L_c}=\frac{v}{2L_o}\Rightarrow L_c=\frac{L_o}{2}$, ratio 1:2.
Question 3 of 12
The first overtone of a closed pipe has the same frequency as the fundamental of an open pipe. The ratio of their lengths (closed:open) is:
1:1
2:3
3:2
3:4
Explanation: Closed first overtone $=\frac{3v}{4L_c}$; open fundamental $=\frac{v}{2L_o}$. Equal $\Rightarrow\frac{3}{4L_c}=\frac{1}{2L_o}\Rightarrow L_c=\frac{3L_o}{2}$, ratio 3:2.
Question 4 of 12
Two strings of the same material and length have radii in ratio 1:2 under the same tension. The ratio of their fundamental frequencies is:
1:2
2:1
1:4
4:1
Explanation: $\mu\propto r^2$ and $f\propto\frac{1}{\sqrt{\mu}}\propto\frac{1}{r}$, so $f_1:f_2=r_2:r_1=2:1$.
Question 5 of 12
A tuning fork gives 4 beats/s with a 320 Hz fork. After filing the tuning fork, the beats become 6/s. The original frequency of the filed fork was:
316 Hz
324 Hz
326 Hz
314 Hz
Explanation: Filing raises a fork's frequency. Beats increased, so the fork was below 320 (316 Hz); filing raised it further from 320 (316→above), increasing beats.
Question 6 of 12
A wave $y=A\sin(\omega t-kx)$ meets a rigid wall and reflects. The reflected wave is:
$A\sin(\omega t+kx)$
$-A\sin(\omega t+kx)$
$A\cos(\omega t+kx)$
$A\sin(\omega t-kx)$
Explanation: Reflection at a rigid boundary reverses direction and adds a phase change of $\pi$, giving $-A\sin(\omega t+kx)$.
Question 7 of 12
A source and an observer both move towards each other at speed $u$ ($v=$ sound speed). The observed frequency is:
$f\frac{v}{v}$
$f\frac{v+u}{v-u}$
$f\frac{v-u}{v+u}$
$f\frac{v}{v-u}$
Explanation: Observer towards (+$u$ in numerator), source towards ($-u$ in denominator): $f'=f\frac{v+u}{v-u}$.
Question 8 of 12
A car moving at $v_s$ towards a wall sounds a horn of frequency $f$. The beat frequency between the direct horn and the echo heard by the driver is largest when $v_s$ is:
zero
small but nonzero
equal to sound speed
negative
Explanation: Beats arise only when the car moves; the echo is Doppler-shifted, so a nonzero (and increasing) $v_s$ gives audible, growing beats.
Question 9 of 12
If the speed of sound increases with temperature, the fundamental frequency of an open pipe on a hot day will:
increase
decrease
stay the same
become zero
Explanation: $f_1=\frac{v}{2L}$; a higher $v$ (warmer air) raises the fundamental frequency.
Question 10 of 12
The number of beats heard per second when 500 Hz and 506 Hz are sounded together, and the maximum loudness occurs at intervals of:
6 beats/s, every $\frac{1}{6}$ s
3 beats/s, every $\frac{1}{3}$ s
6 beats/s, every 6 s
12 beats/s, every $\frac{1}{12}$ s
Explanation: Beat frequency $=|506-500|=6$ Hz, so maxima recur every $\frac{1}{6}$ s.
Question 11 of 12
In a stationary wave on a string, the energy:
flows steadily along the string
is transported only at antinodes
is not transported but remains confined
flows from antinodes to nodes
Explanation: A standing wave transports no net energy; it remains confined between the fixed boundaries.
Question 12 of 12
A whistle of frequency $f$ is whirled in a horizontal circle. A distant observer hears a frequency that:
is constant at $f$
varies between a maximum and a minimum about $f$
only increases
is always higher than $f$
Explanation: As the source alternately approaches and recedes along the line of sight, the Doppler-shifted frequency oscillates above and below $f$.