Lecture 32 Transverse and longitudinal Waves Waves on a wire Guitar strings Waves travelling down a wire or a rope are transverse waves and their wavespeed increases the greater the tension in the wire and reduces as the mass of the string or wire increases The precise formula is F v 1 2 1 Travelling waves are like water waves that move in a particular direction A different sort of wave occurs on guitar string because both ends of the string are fixed The vibrations of the string do not seem to move with the wavespeed v Why not These waves seem to stay in one place and for this reason are called standing waves They can be thought of as a sum of travelling waves where each of the travelling waves in the string moves with the wavespeed v For standing waves to occur there needs to be special relationships between the wavelength of the wave and the length of the string In the case of a guitar the wave amplitude must be zero at each end of the string so that an integral number of half wavelengths must fit in the string length L We must then have 2L 1 n n L or n 2 n 2 The vibrations of the string cause a pressure wave in the air and this pressure wave is the sound which we hear as we will return to later Notice that waves on a guitar string are transverse standing waves The fundamental wavelength is 1 2L The frequency associated with this fundamental f1 v 1 The frequencies associated with the other wavelengths are given by v n fn v nf1 3 n 2L Sound Waves Applications Sound waves are longitudinal waves and have different names depending on the frequency of the waves Human Audible frequencies lie in the range 20Hz f 20 000Hz Infrasonic waves lie below the audible frequency range while ultrasonic waves have frequencies above the audible Ultrasonic 1 waves are used extensively in biology ranging from low intensity applications in imaging e g ultrasound to high intensity applications in surgery e g CUSA A piezoelectric device can be used to generate sound waves as the piezo material oscillates at the same frequency as an applied oscillatory voltage The percentage of the sound wave that is reflected is given by PR h l 2 100 h l 4 Contrast is stongest when the differences in density are largest The speed of sound The speed of sound in a material in a liquid or a gas is given by B v 1 2 5 where B is the bulk modulus and is the density In a solid the sound velocity is given by Y 6 v 1 2 where Y is the Young s modulus Sound travels faster in water or a solid than in air The speed of sound is temperature dependent and in air the dependence is approximated by in meters s v 331 T 1 2 273K 7 Intensity I Intensity is equal to power per unit area and has the units W m2 We therefore have 1 E P ower 8 I Area A t The limits of human hearing at 1kHz are 10 12 W m2 I 1W m2 9 where the lower limit is the faintest sounds that can be heard while the upper limit is the pain threshold Acoustic surgery is possible at very high intensities which requires careful focusing of the acoustic waves Typical 2 surgery intensities are of order Isurgical 107 W m2 At these intensities the cell wall breaks and cells fragment Even medical ultrasound is at intensities of order 100W m2 however the frequency used is well outside the human audible range so it is assumed that no pain is caused by the procedure Due to the broad range of sound levels which are relevant to physiology and technology sound level is usually quoted using a logarithmic scale the decibel scale in a similar way earthquake magnitudes are on a logarithmic scale A sound wave of intensity I in W m2 has a decibel value given by 10Log I I0 10 where I0 10 12 W m2 The threshold of hearing has 0 and the threshold of pain 120 Often sound is emitted from a point source and the intensity at some distance r from the point source is given by I r P 4 r 2 11 We have already seen this sort of equation when we looked at the power emitted by the sun using the Poisson Boltzmann law though we did not use the word intensity However the use of intensity makes thinking about problems like that somewhat easier 3
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