Physics 203 – College Physics I Department of Physics – The Citadel Physics 203 College Physics I Fall 2012 S. A. Yost Chapters 11 – 12 Waves and Sound

Physics 203 – College Physics I Department of Physics – The Citadel Announcements We will discuss Waves and Sound today: Sec. 7 – 9 in Chapter 11 and sec. 1 – 4 and 7 in Chapter 12. Homework set HW12 on this is due Thursday. It is relatively short, taking into account that it is due in 2 days. You will need to read some topics before we discuss them in class. The Final Exam is next Monday: 8 – 11 AM here. Bring a page of notes. I won’t provide equations. I will provide constants, conversion factors, and moments of inertia.

Physics 203 – College Physics I Department of Physics – The Citadel Clicker Question Are you here? A = Yes B = No C = Both D = Neither E = Can’t be determined

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion A disturbance that moves through the medium is called a traveling wave. A traveling wave could be continuous, or just a one-time pulse as shown here: v

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion A pulse could be created by pulling up sharply on one end of a rope. Nearby parts of the rope are pulled up a little later, and the pulse moves down the rope. v

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion Continuously moving the end up and down will make a continuous wave, or periodic wave.

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The distance between peaks is the wavelength .

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The height of the peaks is called the amplitude A. A

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The crests of the wave move with wave velocity v down the rope. v

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The material of the rope doesn’t travel – it just vibrates up and down, while the wave moves through it.

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The time for the wave to travel one wavelength is called the period T.

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion = v T. The wavelength, period, and velocity are related:

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The frequency is the number of crests per unit time that pass a point: seconds

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion In this case, 5 crests passed in 10 seconds, so the frequency is f = 5/10 s = 0.5 s -1

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The frequency is measured in units called Hertz: 1 Hz = 1 s -1 This example had a frequency of 0.5 Hz.

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion If 5 cycles pass in 10 seconds, each cycle lasted T = 2 s. This is the period. In general, T = 1/f.

Physics 203 – College Physics I Department of Physics – The Citadel Wave Motion The wave velocity is the product of the frequency and wavelength: v = /T = f.

Physics 203 – College Physics I Department of Physics – The Citadel Wave Speed The stronger the restoring force when a displacement is made from equilibrium, the faster the wave travels. Waves move slower in a denser material, because more massive materials have more inertia, and resist propagating the wave. Wave on a string of linear mass density  = m/L under tension F T : v = √ F T / 

Physics 203 – College Physics I Department of Physics – The Citadel Standing Waves If a string is tied down at both ends, it can have a standing wave, which doesn’t go anywhere, but vibrates up and down in place with the ends fixed. L There has to be an integer number of half- wavelengths on the string, so = 2L/n The mode above, n = 1, is called the fundamental mode.

Physics 203 – College Physics I Department of Physics – The Citadel Standing Waves The next mode is n = 2. This is called the second harmonic. A full wavelength fits between the ends. L The relation f = v still holds, with v determined by the tension of the string and its mass… The mode above has twice the frequency of the fundamental.

Physics 203 – College Physics I Department of Physics – The Citadel Piano String Example A 1.6 meter steel piano wire with a mass of 38.4 g put under a tension of 1200 N. What is the fundamental frequency heard when the string is struck?

Physics 203 – College Physics I Department of Physics – The Citadel Piano String Example The fundamental frequency can be found from the condition L = /2 where L = 1.63 m is the length of the stretched piano string and is the wavelength of a standing wave with nodes at the end of the string. Then f = v/, where v is the wave speed on the string. The vibrating string will cause he air to vibrate at the same frequency.

Physics 203 – College Physics I Department of Physics – The Citadel Piano String Example To find the wave speed v, we need the relation v = √ F T /  where F T = 1200 N is the tension on the string and  is the linear mass density of the steel spring. What is  ?

Physics 203 – College Physics I Department of Physics – The Citadel Piano String Example The linear mass density of the piano wire is  = m/L = kg/1.6 m = kg/m Since F T = 1200 N and  = kg/m v = √ F T /  =  224 m/s  The frequency is then f = v/ = 224 m/s / (2L) = (224 m/s) / (3.2 m) = 70 Hz.

Physics 203 – College Physics I Department of Physics – The Citadel Sound Waves But how do we hear the piano string? The vibrations must move from the string into the nearby air. Vibrations of air molecules carry the wave to our ears – we call this a sound wave.

Physics 203 – College Physics I Department of Physics – The Citadel Sound Waves Sound waves are longitudinal pressure waves where the air molecules periodically get closer together and further apart, causing variations in the air pressure.

Physics 203 – College Physics I Department of Physics – The Citadel Sound Waves v = f is different for sound waves and the vibrating object. For sound waves v = 343 m/s at ordinary pressures and temperatures (20 o C). For our piano wire, v was 224 m/s.

Physics 203 – College Physics I Department of Physics – The Citadel Sound Waves Since each vibration of the object creates one vibration of the air, the frequencies (vibrations per second) are the same when a wave passes from one media to another.

Physics 203 – College Physics I Department of Physics – The Citadel Piano String Example What is the wavelength of the sound wave produced by this piano string? The frequency in air is also f = 70 Hz since the air vibrates together with the string. The speed of sound in air is v s = 343 m/s. Then the wavelength of the sound waves is s = v s /f = 4.9 m. Greater than the wavelength on the string

Physics 203 – College Physics I Department of Physics – The Citadel Energy in a Spherical Wave If energy carried by a spherical wave is conserved, then the power flowing through any sphere of radius R is equal to the power radiated by the source, assuming no loss. R Source All the power has to flow through this sphere unless there is absorption on the way.

Physics 203 – College Physics I Department of Physics – The Citadel Energy in a Spherical Wave Intensity is power per unit area: I = P/A. The area of a sphere of radius R is A = 4  R 2. Energy conservation: P = 4  R 2 I at any radius R. R Source

Physics 203 – College Physics I Department of Physics – The Citadel Energy in a Spherical Wave The intensities I 1 and I 2 at distances R 1 and R 0 from the source are related by an inverse- square law: I 2 /I 1 = (R 1 /R 2 ) 2. R2R2 Source inverse square law for wave intensity. R1R1

Physics 203 – College Physics I Department of Physics – The Citadel Loudness Our ears can perceive a very wide range of sound intensities. The threshold of hearing is the quietest sound a normal person can hear. This depends on frequency, but is about I 0 = W/m 2 (at a frequency of 1000 Hz).

Physics 203 – College Physics I Department of Physics – The Citadel Loudness I 0 = W/m 2 is taken to be a standard measure of the minimum audible sound intensity. The loudest sounds we can hear (without extreme pain and excessive ear damage) is about 1 W/m 2. This means we can hear 12 orders of magnitude (powers of 10) in variations of sound intensity! Such a wide range of intensity is best represented by a logarithmic scale, which counts powers of 10.

Physics 203 – College Physics I Department of Physics – The Citadel Decibel Scale The sound level of an intensity I in decibels (dB) is given by  = 10 log 10 ( I/I 0 ) (  is used for decibels in Giancoli, but is not necessarily a standard notation.) I 0 = W/m 2 corresponds to 0 dB, I max = 1 W/m 2 corresponds to 120 dB.

Physics 203 – College Physics I Department of Physics – The Citadel Logarithms Some useful properties of logarithms: log 10 (10 N ) = N For any base, log (1) = 0 log (xy) = log x + log y log (x/y) = log x – log y

Physics 203 – College Physics I Department of Physics – The Citadel Example How loud is a drum if the intensity is measured to be I = 3.2 x W/m 2 at a distance R = 5 m from the drum? The sound level in decibels is  = 10 log 10 ( I/I 0 ) = 10 log 10 (3.2 x ) = 105 dB

Physics 203 – College Physics I Department of Physics – The Citadel Decibel Scale When comparing two intensities, the decibel levels are related by     = 10 log 10 ( I 2 /I 1 ) This is useful when comparing any two loudness levels, without having to use the threshold of hearing.

Physics 203 – College Physics I Department of Physics – The Citadel Example How loud would the drum be at a distance of 10 m (twice as far)? The intensity would be ¼ as great, due to the inverse square law for the intensity of spherical waves. (Power = Intensity x area = constant) This would reduce the loudness by  = 10 log 10 (1/4) = dB

Physics 203 – College Physics I Department of Physics – The Citadel Example The loudness of the drum at 10 m would then be 105 dB – 6.0 dB = 99 dB. Note that a factor of 4 in intensity is 6 dB, and a factor of 2 in intensity is 3 dB.

Physics 203 – College Physics I Department of Physics – The Citadel Wind Instruments Wind Instruments produce sounds using a vibrating column of air. For example, consider this tube, open on the ends.

Physics 203 – College Physics I Department of Physics – The Citadel Wind Instruments Air can vibrate back and forth, but at the ends, the pressure must be the same as it is outside the tube. The air can blow in and out freely, keeping the pressure fixed. So there is no pressure variation at the ends.

Physics 203 – College Physics I Department of Physics – The Citadel antinode Wind Instruments Drawing a red line for the difference between the atmospheric pressure and pressure in the pipe, the wave’s pressure maxima and minima would look like this: L node

Physics 203 – College Physics I Department of Physics – The Citadel antinode Wind Instruments Then the frequencies produced by the open tube are f N = Nv/2L = Nf 1 A tube that is closed on one end behaves differently. Read about that case in the book before doing the homework. L node