Question 1
A 2.50 kg string is stretched to 20.0 m under 200 N tension. How long does a transverse disturbance take to reach the other end?
Steps only
μ=0.125 kg/m; v=√(200/0.125)=40 m/s; t=20/40=0.50 s
Answer
0.50 s
CLASS 11 · PHYSICS · CHAPTER 14
Waves
Simple worked answers to all 19 end exercises on wave speed, sound, standing waves, resonance and beats.
19 worked answersComplete chapter answers
19 worked answers · short steps first, explanations when you need them
Chapter 14 Exercises
Textbook pages 298–300
Question 2
A stone falls 300 m into a pond. When is the splash heard at the top? Take sound speed as 340 m/s and g=9.8 m/s².
Steps only
t=√(2×300/9.8)+300/340≈7.82+0.88=8.70 s
Answer
8.70 s
Question 3
A 12.0 m steel wire has mass 2.10 kg. Find the tension needed for a transverse wave speed of 343 m/s.
Steps only
T=μv²=(2.10/12.0)(343)²≈2.06×10⁴ N
Answer
2.06×10⁴ N
Question 4
Using v=√(γP/ρ), explain why sound speed in air (a) is independent of pressure, (b) increases with temperature, and (c) increases with humidity.
Steps only
Use v=√(γRT/M): pressure cancels; higher T and lower molar mass in humid air both raise v.
Answer
Pressure: no change; temperature: increases; humidity: increases.
Question 5
Can each function represent a travelling wave: (a) (x−vt)², (b) log[(x+vt)/x₀], (c) 1/(x+vt)?
Steps only
Each is a function of x−vt or x+vt, so each can travel; note the singular point for (c).
Answer
All three, where defined.
Question 6
A 1000 kHz bat sound meets a water surface. Find the wavelengths of (a) reflected sound in air and (b) transmitted sound in water. Speeds: 340 m/s in air and 1486 m/s in water.
Steps only
λair=340/10⁶=3.40×10⁻⁴ m; λwater=1486/10⁶=1.486×10⁻³ m
Answer
(a) 0.340 mm; (b) 1.486 mm
Question 7
Find the wavelength in tissue if sound speed is 1.7 km/s and scanner frequency is 4.2 MHz.
Steps only
λ=v/f=1700/(4.2×10⁶)=4.05×10⁻⁴ m
Answer
0.405 mm
Question 8
For y(x,t)=3.0 sin(36t+0.018x+π/4), with x and y in cm and t in s, state wave type, direction, speed, amplitude, frequency, initial phase and wavelength.
Steps only
kx+ωt ⇒ −x; v=36/0.018=2000 cm/s; f=36/2π; λ=2π/0.018 cm
Answer
−x; 20 m/s; 3 cm; 5.73 Hz; π/4; 3.49 m
Question 9
For the wave in Exercise 14.8, write y(t) at x=0, 2 and 4 cm. What differs from point to point?
Steps only
Substitute x=0, 2, 4 cm into the wave equation. The three time graphs are sinusoids with equal amplitude and frequency, but different phase.
Answer
Same amplitude and frequency; phase differs.
Question 10
For y=2.0cos 2π(10t−0.0080x+0.35), where x and y are in cm and t in s, find phase differences for separations (a) 4 m, (b) 0.5 m, (c) λ/2, (d) 3λ/4.
Steps only
Δφ=2π(0.0080)Δx(cm); also Δφ=2πΔx/λ
Answer
(a) 6.4π; (b) 0.8π; (c) π; (d) 3π/2 rad
Question 11
A 1.5 m string of mass 3.0×10⁻² kg has y=0.06 sin[(2π/3)x]cos(120πt), with x,y in metres. Identify the wave, and find the component waves' wavelength, frequency, speed and string tension.
Steps only
k=2π/3 ⇒ λ=3 m; ω=120π ⇒ f=60 Hz; v=180 m/s; μ=0.02 kg/m ⇒ T=μv²=648 N
Answer
Standing; λ=3 m; f=60 Hz; v=180 m/s; T=648 N
Question 12
For the standing wave in Exercise 14.11, compare the frequency, phase and amplitude at different points. Find the amplitude 0.375 m from an end.
Steps only
A(x)=0.06|sin(2πx/3)|; at x=0.375 m, A=0.06sin(π/4)=0.0424 m
Answer
60 Hz; phase depends on loop; amplitude≈4.24 cm
Question 13
Classify each as travelling, stationary or neither: (a) 2cos(3x)sin(10t), (b) (x−vt)², (c) 3sin(5x−0.5t)+4cos(5x−0.5t), (d) cos x sin t+cos 2x sin 2t.
Steps only
Separate position/time factors for standing waves; check for a single x±vt phase for travelling waves.
Answer
(a) stationary; (b), (c) travelling; (d) neither
Question 14
A wire fixed at both ends vibrates in its fundamental mode at 45 Hz. Its mass is 3.5×10⁻² kg and linear density is 4.0×10⁻² kg/m. Find wave speed and tension.
Steps only
L=m/μ=0.875 m; v=2Lf=78.75 m/s; T=μv²≈248 N
Answer
78.8 m/s; 248 N
Question 15
A 1 m tube closed at one end resonates with a 340 Hz tuning fork at lengths 25.5 cm and 79.3 cm. Estimate sound speed, ignoring end effects.
Steps only
λ=2(0.793−0.255)=1.076 m; v=340×1.076≈366 m/s
Answer
366 m/s
Question 16
A 100 cm steel rod clamped at its middle has fundamental longitudinal frequency 2.53 kHz. Find sound speed in steel.
Steps only
λ=4(L/2)=2L=2 m; v=fλ=2.53×10³×2=5.06×10³ m/s
Answer
5.06×10³ m/s
Question 17
A 20 cm pipe is closed at one end. Which mode resonates with 430 Hz? Will it resonate if both ends are open? Use sound speed 340 m/s.
Steps only
Closed: f₁=v/4L=425 Hz≈430 Hz. Open: f₁=v/2L=850 Hz; 430 Hz is not a harmonic.
Answer
Closed pipe: fundamental; open pipe: no resonance.
Question 18
Two sitar strings make 6 Hz beats. Slightly reducing string A's tension reduces the beats to 3 Hz. If A was 324 Hz, find B's frequency.
Steps only
A's frequency falls; since the beat rate also falls, B must be above A: fB=324+6.
Answer
330 Hz
Question 19
Explain: (a) displacement nodes are pressure antinodes in sound, (b) how bats locate obstacles, (c) why same-frequency violin and sitar notes sound different, (d) why solids support transverse and longitudinal waves but gases only longitudinal waves, (e) why a pulse changes shape in a dispersive medium.
Steps only
Pressure variation tracks the displacement gradient; echoes locate objects; overtones set timbre; shear elasticity allows transverse waves; dispersion separates frequencies.
Answer
(a)–(e) follow from displacement gradients, echoes, overtones, shear elasticity and frequency-dependent wave speed.