# Physics 131: tutorial week 6 Waves (1)

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1 Physics 131: tutorial week 6 Waves (1) Take the speed of sound in air at 0 C as 331ms 1 and the speed of electromagnetic radiation c 3, ms A person standing in the ocean notices that after a wave crest passes by, ten more crests pass in a time of 120s. What is the frequency of the wave? (0,083Hz) Each wave goes by in a time equal to the period of the wave, hence 10 T 120 s and f 1 T 1 0, 083 Hz A light wave travels through air at a speed of 3, ms 1. Red light has a wavelength of about 6, m. What is the frequency of red light? (4, Hz) f v 3, λ 6, , Hz. 3. The right-most key on a piano produces a sound wave that has a frequency of 4185,6Hz. Assuming the speed of sound in air is 343ms 1, find the corresponding wavelength. (8, m) λ v f 343 0, 0819 m. 4185, 6 4. One of the Radio Zulu broadcasting frequencies is 90,8MHz. What is the corresponding wavelength? (3,30m) λ v 3, , 30 m. f 90, A person fishing from a pier observes that four wave crests pass by in 7,0s and estimates the distance between two successive crests as 4,0 m. The timing starts with the first crest and ends with the fourth. What is the speed of the wave? (1,7ms 1 ) The number of complete waves that pass by is three, hence T 3 0, 43 s. 7, 0 The estimated distance of 4,0m between crests is the wavelength, so v fλ 0, 43 4, 0 1, 7 ms The wavelength of a sound wave in air is 2,76m at 0 C. What is the wavelength of this sound in water at 0 C? (Use the data on p 3 of your notes.) (Hint: since the frequency is dependent on the source, it is the same in both media.) (12,3m) Since the frequency is dependent on the source, it is the same in both air and water. f v air v water. λ air λ water Hence λ water λ air vwater v air 2, , 3 m.

2 7. At a height of ten metres above the surface of a lake, a sound pulse is generated. The echo from the bottom of the lake returns to the point of origin 0,140s later. The air and water temperatures are 0 C. How deep is the lake? (Refer to the table on page 3 of your notes.) (59m) The distance travelled by the pulse in air is m. Let d be the depth of the lake, then the distance travelled by the pulse in water is 2d. If the time the pulse travels in air is t a, then the time the pulse travels through the water is (0, 140 t a ). Since waves travel at constant speed in a medium, we can use the relation s vt. Hence In air : t a. In water : 2h 1480 (0, 140 t a ). ( Hence h , ) 59 m On a day when the temperature is 0 C, a man is watching as spikes are being driven to hold a steel rail in place. The sound of each sledgehammer blow reaches him in 0,14s through the rail and in 2,0s through the air, after he sees the blow fall. Find the speed of sound in the rail. (4, ms 1 ) The distance travelled by the sound is the same in air and through the rail. Hence s v air t air v rail t rail , 0 Hence v rail 4, ms 1. 0, If the speed of sound in air at 0 C is 331ms 1, what will it be at 27 C? (347ms 1 ) T T where T is in degrees celsius Hence ms Calculate (i) the speed (ii) the wavelength of a sound wave in air at 45 C produced by a tuning fork of frequency 400Hz. (357ms 1 ; 0,893m) (i) ms (ii) λ f 357 0, 893 m Standing waves are set up on a string with a frequency of 20Hz and a distance between succesive nodes of 0,070 m. Determine the speed with which the separate waves travel along the string. (2,8ms 1 ) The nodes of a standing wave are separated by a distance of half the wavelength, so λ 2 0, 070 0, 14m. v f λ 20 0, 14 2, 8 ms 1.

3 12. With a source of sound of frequency 5000 Hz, stationary waves are formed in air at 0 C. The distance between successive antinodes is 3,31cm. At a higher temperature it is found that the separation of the antinodes with the same source is 3,43cm. Calculate (i) the speed of sound in air at 0 C (ii) the temperature at which the second observation was made. (331ms 1 ; 20,2 C) (i) The wavelength is obtained by taking twice the distance between the antinodes. fλ , 31 3, ms 1. (ii) T ( ) 2 ( ) Hence T , 2 C A tuning fork vibrates at a frequency of 521Hz. An out-of-tune piano string vibrates at 519Hz. How much time separates successive beats? (0,5s) Beat frequency f 1 f Hz, hence T 1 f 0, 5 s. 14. Two guitars are slightly out of tune. When they play the same note simultaneously, the sounds they produce have wavelengths of 0,776 m and 0,769 m. On a day when the speed of sound is 343ms 1, what beat frequency is heard? (4Hz) Beat frequency f 2 f 1 λ 2 λ , , Hz. 15. A source of frequency 256Hz is emitting sound energy in air at 0 C. The velocity of sound in air at 0 C is 330ms 1. (i) If an observer sets up a stationary wave system by intercepting the original sound wave with a reflecting barrier, what would be the distance between successive nodes? (ii) If the air temperature rises to 20 C, what would be the new sound wave velocity? (iii) If the observer moves steadily towards the surce, would the apparent frequency be higher than, the same as, or lower than the true frequency? Explain your answer carefully but without the use of formulae. (iv) What beat frequency would be heard if a second source is sounded, of frequency 260Hz (the observer being stationary)? (0,645m; 342ms 1 ; higher; 4Hz) 1 (i) We require the distance between nodes: 2 λ 1 2 f 330 0, 645 m T 293 (ii) ms 1. (iii) The observer encounters more crests and troughs per unit time when moving towards the source, hence the frequency would be higher. (iv) f f 2 f Hz.

4 16. At a football game, a stationary spectator is watching the halftime show. A trumpet player in the band is playing a 784Hz tone while marching directly towards the spectator at a speed of 0,900ms 1. On a day when the speed of sound is 343ms 1, what frequency does the spectator hear? (786Hz) The source is moving towards the listener so the speed must be taken as positive, hence f v s f Hz , A hawk is flying directly away from a bird watcher at a speed of 11,0ms 1. The hawk produces a shrill cry whose frequency is 865Hz. The speed of sound is 343ms 1. What is the frequency that the bird watcher hears? (838Hz) The source is moving away from the listener so the speed must be taken as negative, hence f v s f Hz , On a day when the speed of sound in air is 340,0ms 1 a siren mounted on a car emits a note whose frequency is 1000Hz. (i) Determine the frequency of the sound heard by a stationary observer when the car approaches him with a speed of 40,00ms 1. (ii) Determine the frequency heard by an observer moving toward the car with a speed of 40,00ms 1 while the car remains stationary. (1133Hz; 1118Hz) (i) f 340, 0 f Hz. v s 340, 0 40, 00 (ii) f + v o 340, , 00 f Hz. 340, A speeding motor-cyclist passes a stationary police car. The car siren is started (the car still being stationary) and the motor-cyclist hears a frequency of 900Hz. If the true siren frequency is 1000Hz and the speed of sound in the air is 343ms 1, (i) calculate the speed of the motor-cyclist. The police car starts up in pursuit and soon reaches a speed of 43,0ms 1, still sounding its siren. (ii) what frequency does the motor-cyclist no hear? (34,3ms 1 ; 1029Hz) (i) The observer, being the motor-cyclist, is moving away from the source, so we must expect the calculated velocity to be negative. f + v o f, ( ) f hence v o f 1 ( ) , 3 ms 1. (ii) Both the source and the observer are now moving. The observer is still moving away from the source, whilst the source is moving towards the observer. The motor-cyclist s speed must therefore be taken as negative and the police car s speed as positive. f + v o v s f , Hz , 0

5 20. The security alarm on a parked car goes off and produces a frequency of 1000Hz. As you drive towards this parked car, pass it and drive away, you observe the frequency to change by 99 Hz. Calculate your own speed. (Take the speed of sound in the air as 343ms 1.) (17ms 1 ) For this problem it is convenient to change the formula for the doppler effect to reflect the direction of motion of the observer and take the absolute value of the observer s speed. We denote the apparent frequencies when moving towards and away from the source as f t and f a respectively. Thus f t + v o f and f a v o f. We know that the difference f t f a 99Hz (since f t > f a). Now ( + f t f a vo v ) o f 2v o f, hence v o (f t f a ) 2f ms A tuning fork of frequency 400Hz is moved away from an observer and towards a flat wall with a speed of 2ms 1, the air temperature being 27 C. What is the apparent frequency (i) of the unreflected sound wave coming directly towards the observer and (ii) of the sound waves coming to the observer after reflection? (iii) How many beats are heard? (398 Hz; 402 Hz; 4 Hz) (i) The source is moving away from the observer so f v s f where v s 2 ms 1 and 331 Hence f Hz (ii) The source is now moving towards the observer, so f Hz (iii) Beat frequency Hz ms On a hot night a bat emitting squeaks of frequency 100kHz flies with a speed of 20ms 1 towards a stationary bat which is emitting squeaks of the same frequency. When the squeaks of the two bats occur together, a beat note of frequency 6kHz is produced, as recorded by a suitable detector near the stationary bat. Calculate the night temperature. (Take the speed of sound in air at 0 C as 330ms 1.) (40 C ) The recorder is the observer which hears f f v s Hz. Beat frequency f f Hz. Solving the above gives 353, 3ms 1. The temperature is calculated from 0 T T. Thus 273 ( ) 2 353, C. 330

6 True or false questions TRUE FALSE 1. The expression v fλ applies to both transverse waves and also to longitudonal ones. 2. When a sound wave passes from air into water, the same number of waves per second must enter the water as leave the air. So the frequency stays the same. The wavelength however, increases. 3. The ratio of the speed of sound in air at 35 C to that in air at 15 C is given by Since water is denser than air, sound travels more slowly through it. 5. At the antinodes in a stationary wave system, the resultant displacement is always a maximum. 6. In the doppler effect, the speed of the waves emitted by the source depends on the speed of the source. 7. The Doppler shift for a source moving towards a stationary observer with velocity v is not the same as for an observer moving towards a stationary source with velocity v.

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