PHYSICS 231 Lecture 35: interference & sound Remco Zegers Question hours:Monday 9:15-10:15 Helproom PHY 231

example A pendulum with a length of 4 m and a swinging mass of 1 kg oscillates with an maximum angle of 10o. What is the gravitational force parallel to the string, perpendicular to the string, the total gravitational force and the centripetal force when the mass passes through the equilibrium position and when it reaches its maximum amplitude? equilibrium maximum ampl. gravitation // to string 1gcos(00)=9.8N 1gcos(100)=9.65N perpendicular 1gsin(00)=0N 1gsin(100)=1.7N total gravitational 9.8N 9.8N (vectors!!) centripetal 0.3N (mv2/L) 0 N PHY 231

describing a traveling wave : wavelength distance between two maxima. While the wave has traveled one wavelength, each point on the rope has made one period of oscillation. v=x/t=/T= f On a string: v=(F/) PHY 231

A person is tuning a guitar string. He makes the tension quiz (extra credit) A person is tuning a guitar string. He makes the tension in the string 4 times larger than it originally was. If the wavelength of the oscillations through the string remains constant, by what factor does the frequency of the produced sound change? ¼ ½ 1 2 4 v=(F/) if Fx4 then vx2 v=x/t=/T= f if vx2 then fx2 if =constant PHY 231

Interference Two traveling waves pass through each other without affecting each other. The resulting displacement is the superposition of the two individual waves. example: two pulses on a string that meet PHY 231

Interference II constructive interference destructive interference PHY 231

+ = + = Interference III   demo: interference destructive interference waves ½ out of phase + = + = constructive interference waves in phase PHY 231

Interference IV If the two interfering Two interfering waves can waves always have the same vertical displacement at any point along the waves, but are of opposite sign: standing waves Two interfering waves can at times constructively interfere and at times destructively interfere later more!!! PHY 231

Interference holds for any wave type The pulses can be sine-waves, rectangular waves or triangular waves PHY 231

Interference in spherical waves if r2-r1=n then constructive interference occurs if r2-r1=(n+½) the destructive interference occurs r1 r2 r1=r2 maximum of wave minimum of wave positive constructive interference destructive interference negative constructive interference PHY 231

Interference of water waves PHY 231

Example two speakers separated by 0.7m produce a sound with frequency 690 Hz (from the same sound system). A person starts walking from one of the speakers perpendicular to the line connecting the speakers. After what distance does he reach the first maximum? And the first minimum? vsound=343 m/s d1 d2 v=f so =v/f=343/690=0.5m 1st maximum: d1-d2=1=0.5 d2=(0.72+d12) d1-(0.72+d12)=0.5 d1=(-)0.24m 1st minimum: d1-d2=½=0.25 d1-(0.72+d12)=0.25 d1=(-)0.855 0.7m direction of walking person PHY 231

Reflection of waves. Frope on wall= -Fwall on rope demo: rope on wall FREE END: no inversion Frope on wall= -Fwall on rope demo: rope on wall FIXED END: pulse inversion PHY 231

Connecting ropes If a pulse travels from a light rope to a heavy rope (light< heavy) the boundary is nearly fixed. The pulse is partially reflected (inverted) and partially transmitted. If a pulse travels from a heavy rope to a light rope (light< heavy) the boundary is nearly free. The pulse is partially reflected (not inverted) and partially transmitted. before before Ain Ain AR AT AR AT after |AR|<|Ain| |AT|<|Ain| after |AR|<|Ain| |AT|>|Ain| PHY 231

Sound: longitudinal waves PHY 231

The speed of sound Depends on the how easy the material is compressed (elastic property) and how much the material resists acceleration (inertial property) v=(elastic property/inertial property) v=(B/) B: bulk modulus : density The velocity also depends on temperature. In air: v=331(T/273 K) so v=343 m/s at room temperature PHY 231

Quick question The speed of sound in air is affected in changes in: (more than one possible) wavelength frequency temperature amplitude none of the above answer c) PHY 231

Intensity Intensity: rate of energy flow through an area Power (P) J/s A (m2) I=P/A (J/m2s=W/m2) example: If you buy a speaker, it gives power output in Watts. However, even if you put a powerful speaker in a large room, the intensity of the sound can be small. PHY 231

Intensity Faintest sound we can hear: I~1x10-12 W/m2 (1000 Hz) Loudest sound we can stand: I~1 W/m2 Factor of 1012? Loudness works logarithmic… PHY 231

decibel level  =10log(I/I0) I0=10-12 W/m2 y=log10x inverse of x=10y (y=ln(x) x=ey) log(ab) =log(a)+log(b) log(a/b) =log(a)-log(b) log(an) =nlog(a) PHY 231

decibels =10log(I/I0) I0=10-12 W/m2 2-1=10 An increase of 10 dB: intensity of the sound is multiplied by a factor of 10. 2-1=10 10=10log(I2/I0)-10log(I1/I0) 10=10log(I2/I1) 1=log(I2/I1) 10=I2/I1 I2=10I1 Next quiz… PHY 231

Frequency vs intensity 1000 Hz PHY 231

example A machine produces sound with a level of 80dB. How many machines can you add before exceeding 100dB? 1 machine 80 dB=10log(I/I0) 8=log(I/I0)=log(I/1E-12) 108=I/1E-12 I1=10-4 W/m2 ?? machines 100 dB=10log(I/I0) 10=log(I/I0)=log(I/1E-12) 1010=I/1E-12 I??=10-2 W/m2 I1/I??=10-4/10-2=1/100 The intensity must increase by a factor of 100; one needs to add 99 machines. PHY 231