Period and Frequency
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Wave unit Problems
SPH 3U
A child on a swing observes the following. It takes her 2.0 minutes to complete 12 complete cycles. The path distance
of one cycle is about 8.5 meters. What is her period & frequency?
2 min = 120 s f = 12/120 = .1hz
T = 1/f (1/.1) = 10s
What is the period of the second hand, the minute hand, & the hour hand on a clock?
T=60s, (60x60) = 3600s, (3600x12hr) 43 200s
What is the frequency of a strobe light if it flashes 158 times in 1.2 minutes? What is the period?
1.2*60 = 72s
158/72 = 2.2 hz
An object completes 240 cycles in 15 s. What is its frequency?
240/15 = 16 hz
Amplitude, Wavelength, Time, Wave Velocity, Frequency
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A transverse wave has amplitude of 2.4 m. What is the vertical distance, in metres, between the top of a crest and the
bottom of a trough? 2.4 (2) = 4.8m
A sound wave with a frequency of 256 Hz, travelling at 340 m/s, has a wavelength of…? λ = 340/256 = 1.3m
What is the speed of a sound wave with a wavelength of 3.4 m and a frequency of 100 Hz? Vw = 3.4(100) = 340 m/s
Wave velocity, tension, linear density
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The thickest string on the world’s largest guitar has a mass of 400.g and a length of 2.75 m, what is its linear density?
400 g = .4 kg linear density = m/l  .4/2.75 = .144 kg/m
If a string on a wave machine has a wave speed of 200.0 m/s and a tension of 250. N, what is the linear density of the
string? 200 = √ 250/μ  2002 = 250/μ 
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250/2002 = μ = .00625 kg/m
If a wave machine string has a linear density of .6 kg/m, and a wave speed of 250m/s, what tension is required?
250 m/s = √ F/.6  2502 (.6) = F = 37500 N
A string on a violin has a linear density of .05 kg/m, and wave speed of 35 m/s, what must the tension of the string be?
35 m/s = √ F/.05  352 (.05) = F = 61.25 N
Speed of sound & Mache
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Calculate the time it would take for sound to travel through 2000 m of the following substances, given the velocity of
sound through the substance: (a) helium – 972 m/s, (b) water – 1450 m/s, (c) steel – 5130 m/s, (d) glass – 4700 m/s;
(2000/972 = 2.06 s, 1.38 s, 0.390 s, 0.426 s)
2. On a hot summer night (32 °C), you are listening to a rock group in a stadium 350 m away. A friend of yours is sitting
in an air-conditioned house across the country, listening to the broadcast on the radio. If the signal (v = 3.0 × 108 m/s)
travels 30000 km up to a satellite that retransmits it to your friend, who hears the concert first? (your friend)
Temp = 331.4 + (.606*32) = 350.8 m/s  1 second away
Friend = 300 000 000 m/s travels 30 000 km or 30 000 000m there and back (60 000 000m total)
3. If the local speed of sound is 365 m/s and a flying pig is traveling at 890 km/h, what is the Mach number?
890 km/hr = 247.2 m/s 247.2/365 = .677
4. How fast is Mach 3? (Assume 0oC and standard atmospheric pressure).
Temp = 331.4 + (.606*0) = 331.4 m/s mach 3 = 3 times speed of sound…. 331.4(3)994.2 m/s
Standing Waves
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A three-loop standing wave is generated in a string by attaching one end to a wall and letting the transmitted and
reflected waves interfere. If the wavelength of the wave is 15 cm, how far from the wall is the first antinode created?
First antinode (crest) will be ¼ of a wave length…. 15*.25 = 3.75
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A standing wave is created in a string of length 12.5 m. If the waves travel at 5.0 m/s with a frequency of 2.0 Hz, how
many loops are present in the string? 5 = λ / (2)
λ = 2.5m
length of one loop = 1/3 lambda 2.5(.5) = 1.25m
1.25
= one node, 12.5/1.25 = 10 loops
Doppler Effect
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A 250 Hz sound source is moving at 30 m/s towards an observer. Assume the speed of sound is 340 m/s. What
frequency will be heard by the observer? (274 Hz)
A frequency of 530 Hz is heard as a sound source moves away from an observer at 22 m/s. The temperature is 20 ˚C.
What is the frequency of the sound source? (564 Hz)
Multi-step problems / Thinking
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On a chilly winter’s day (-20oC), an echo returned in 3.00 s. How far away was the reflecting surface?
Temp = 331.4 + (.606*-20) = 319.28 m/s
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A cliff is located 100 m away from a climber. If the climber shouts, how long will it take the echo to return if air temperature is 30oC?
Temp = 331.4 + (.606*30) = 349.58 m/s
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 3 second return trip = 1.5 seconds one way  319.28*1.5 = 478.92m
 distance round trip = 100(2) = 200m  200/349.58 = .57s
The period of a sound wave emitted by a vibrating guitar string is 3.0 x 10 s. If the speed of the sound wave is 360 m/s what
is its wavelength? f = 1/T = 1/30 = .03  360 = λ (.03)  λ = 360(.03) = 10.8m
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The speed of light through air is 3.00 x 108 m/s. At a temperature of -52.5oC, how many times faster does light travel through
air than does sound? 300 000 000 m/s = air  Temp = 331.4 + (.606*-52.5) = 299.585 m/s or 300 m/s  1 000 000 x
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If a 4.0 m long string on the same wave machine has a tension of 185 N, and the wave speed is 300. m/s, what is the mass of the string?
300 m/s = √ 185/μ
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3002 = 185/μ
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185/3002 =
μ = .002 kg/m
 μ = m/l
.002 = m / 4.0  m=.008kg
A submarine emits a sonar pulse, which returns from an underwater cliff in 1.02 s. If the speed of sound in salt water is 1531
m/s, how far away is the cliff? 1.02s / 2 = .51s per trip  1531m/s * .51s = 780.81m
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A dolphin squeals when a shark is 42.0 m away from it. How long will it take the echo to return from the shark if the speed
of sound through sea water is 1531 m/s? 42m (2) = 84m round trip
84/1531 = .055s
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Calculate the wavelength of sound produced by a 256 Hz tuning fork in 15 oC air. (1.33 m)
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A thunderclap is heard 5 seconds after the lightening-flash has been seen. How far away is the flash if the temperature is 15 oC? (1.7 km)
10. Water waves in a lake travel 4.4 m in 1.8 s. The period of oscillation is 1.2 s. What is the speed and wavelength of the water
waves? 4.4/1.8 = 2.44 m/s
T = 1.2s
f = 1/T = .833 Hz
λ = 2.44*1.2 = 2.933 m
11. A 128 Hz signal has a wavelength of 279 cm. Find the velocity of the signal in air, and the temperature of the air.
(357 m/s, 42 oC)
12. Two people are fishing from small boats located 30 m apart. Waves pass through the water, and each person’s boat bobs up
and down 15 times in 1.0 min. At a time when one boat is on a crest, the other one is in a trough, and there is one crest
between the two boats. What is the speed of the water? f = 15/60 = .25 Hz
20m = λ
V = λ*f = 20*.25 = 5m/s
13. The wavelength of a water wave is 3.7 m and its period is 1.5s. Calculate
(a) the speed of the wave. 3.7/1.5 = 2.47 m/s
(b) the time required for the water to travel 100m. 100/2.47 = 40.5s
crest to trough 1/2 λ  1.5 λ btw  30/1.5 =
(c) the distance traveled by the wave in 1.0 min. 2.47*60 = 148.2m
14. A group of swimmers is resting in the sun on an off-shore raft, watching the hungry sharks circling them. They estimate that
3.0 m separates a trough and an adjacent crest of surface waves on the lake. They count 14 crests that pass by the raft in 20.0
s. How fast are the waves moving? Trough to crest (3m) = 1/2 λ  λ = 6m
f=14/20 = .7 Hz V=6*.7 = 4.2 m/s
15. The distance between the second and sixth crests in a wave is 75 cm, and one crest travels a distance of 25 cm in 3.0 s. Find
the frequency of the wave. V = 25/3 = 8.3 cm/s λ = 75/4 = 18.75cm f = v/ λ = 8.3/18.75 = .44hz