Chapter 17 The Principle of Linear Superposition and Interference Phenomena

17.1 The Principle of Linear Superposition When the pulses merge, the Slinky assumes a shape that is the sum of the shapes of the individual pulses.

17.1 The Principle of Linear Superposition When the pulses merge, the Slinky assumes a shape that is the sum of the shapes of the individual pulses.

17.1 The Principle of Linear Superposition THE PRINCIPLE OF LINEAR SUPERPOSITION When two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves.

Superposition

Superposition The drawing shows a graph of 2 pulses traveling toward each other at t = 0s. Each pulse has a constant speed of v = 1 cm/s. When t = 2.0, what is the height of the resultant pulse at a) x = 3 cm, and b) x = 4 cm

Superposition The drawing shows a graph of 2 pulses traveling toward each other at t = 0s. Each pulse has a constant speed of v = 1 cm/s. When t = 2.0, what is the height of the resultant pulse at a) x = 3 cm, and b) x = 4 cm. Ans: a) -3cm, b)-2 cm

17.2 Constructive and Destructive Interference of Sound Waves When two waves always meet condensation-to-condensation and rarefaction-to-rarefaction, they are said to be exactly in phase and to exhibit constructive interference.

17.2 Constructive and Destructive Interference of Sound Waves When two waves always meet condensation-to-rarefaction, they are said to be exactly out of phase and to exhibit destructive interference.

17.2 Constructive and Destructive Interference of Sound Waves Pilots use noise-canceling headsets like the ones shown below

17.2 Constructive and Destructive Interference of Sound Waves If the wave patters do not shift relative to one another as time passes, the sources are said to be coherent. For two wave sources vibrating in phase, a difference in path lengths that is zero or an integer number (1, 2, 3,.. ) of wavelengths leads to constructive interference; a difference in path lengths that is a half-integer number (½, 1 ½, 2 ½,..) of wavelengths leads to destructive interference.

17.2 Constructive and Destructive Interference of Sound Waves Example 1 What Does a Listener Hear? Two in-phase loudspeakers, A and B, are separated by 3.20 m. A listener is stationed at C, which is 2.40 m in front of speaker B. Both speakers are playing identical 214-Hz tones, and the speed of sound is 343 m/s. Does the listener hear a loud sound, or no sound?

17.2 Constructive and Destructive Interference of Sound Waves Calculate the path length difference. Calculate the wavelength. Because the path length difference is equal to an integer (1) number of wavelengths, there is constructive interference, which means there is a loud sound.

17.2 Constructive and Destructive Interference of Sound Waves Conceptual Example 2 Out-Of-Phase Speakers To make a speaker operate, two wires must be connected between the speaker and the amplifier. To ensure that the diaphragms of the two speakers vibrate in phase, it is necessary to make these connections in exactly the same way. If the wires for one speaker are not connected just as they are for the other, the diaphragms will vibrate out of phase. Suppose in the figures (next slide), the connections are made so that the speaker diaphragms vibrate out of phase, everything else remaining the same. In each case, what kind of interference would result in the overlap point?

17.2 Constructive and Destructive Interference of Sound Waves

17.4 Beats Two overlapping waves with slightly different frequencies gives rise to the phenomena of beats.

17.4 Beats The beat frequency is the difference between the two sound frequencies: f beat = | f 1 - f 2 |

17.5 Transverse Standing Waves String fixed at both ends A standing wave will develop or L = n (λ n /2) Why is that?

17.5 Transverse Standing Waves Since T = 1/f, and v = fλ, this leads to: L = (λ n /2) and a standing wave of one segment. How about standing waves of more than one segment? In reflecting from the wall, a forward-traveling half-cycle (top 2 waves) becomes a backward-traveling half-cycle (bottom two waves) that is inverted. If the timing is just right the backward-traveling reflected wave will constructively interfere with the forward- traveling wave. When does this occur? When the wave period (T) equals the time it takes the wave to travel from the oscillation source to the reflecting point and return: T = 2L/v

17.5 Transverse Standing Waves Transverse standing wave patterns.

For a string of fixed length L, the boundary conditions can be satisfied only if the wavelength has one of the values A standing wave can exist on the string only if its wavelength is one of the values shown above. Because λf = v, the frequency corresponding to wavelength λ m is Standing Waves on a String

n (or m) is the harmonic number. You can tell a string’s harmonic number by counting the number of loops or segments on the string. 1.The fundamental mode, with n = 1, has λ 1 = 2L, not λ 1 = L. Only half of a wavelength is contained between the boundaries, a direct consequence of the fact that the spacing between nodes is λ/2. 2.The frequencies of the normal modes form a series: f 1, 2f 1, 3f 1, …The fundamental frequency f 1 can be found as the difference between the frequencies of any two adjacent modes. That is, f 1 = Δf = f m+1 – f m. Standing Waves on a String

Node spacing on a string

EXAMPLE 21.1 Node spacing on a string

Standing Sound Waves A long, narrow column of air, such as the air in a tube or pipe, can support a longitudinal standing sound wave. It is often useful to think of sound as a pressure wave rather than a displacement wave. The pressure oscillates around its equilibrium value. Musical instruments use longitudinal standing waves in tubes that are either open at both ends (e.g. flute, organ) or open at one end (e.g. clarinet, trumpet). An open end acts as an antinode (maximum vibration) while a closed end acts as a node, just like for a wave on a string.

Tube open at both ends This is the same as for a transverse standing wave on a string, except that the antinodes are at the ends.

The length of an organ pipe QUESTION: v sound = 343 m/s An organ pipe is open at both ends

The length of an organ pipe

Tube open at one end The open end is an antinode and the closed end is a node. The distance between node and antinode is λ/4. Therefore the tube length (L) must be equal to an odd number of quarter wavelengths:

The notes on a clarinet in a tropical country

The notes on a clarinet

Check your understanding A tube is found to have standing waves at 390, 520 and 650 Hz and no frequencies in between. The behavior of the tube at frequencies higher/lower than that is not known. a.Is this an open-open tube or an open- closed tube? b. How long is the tube?

Check your understanding A tube is found to have standing waves at 390, 520 and 650 Hz and no frequencies in between. The behavior of the tube at frequencies higher/lower than that is not known. a.Is this an open-open tube or an open- closed tube? Open-open b. How long is the tube? – 1.32 m