University Physics: Waves and Electricity Ch17. Longitudinal Waves Lecture 4 Dr.-Ing. Erwin Sitompul

4/2 January–April 2010University Physics: Wave and Electricity Homework 3: Standing Waves Two identical waves (except for direction of travel) oscillate through a spring and yield a superposition according to the equation (a)What are the amplitude and speed of the two waves? (b)What is the distance between nodes? (c)What is the transverse speed of a particle of the string at the position x = 1.5 cm when t = 9/8 s?

4/3 January–April 2010University Physics: Wave and Electricity Solution of Homework 3 (a)Identical except direction of travel ► standing waves: (b)Distance between nodes:

4/4 January–April 2010University Physics: Wave and Electricity Solution of Homework 3 (c)Transversal speed: At x = 1.5 cm = 15 mm and t = 9/8 s = 3/160 min,

4/5 January–April 2010University Physics: Wave and Electricity Sound Waves  From previous chapter we know that mechanical waves are classified into transverse waves and longitudinal waves.  In this class, a sound wave is defined roughly as a longitudinal waves.  The figure above illustrates several ideas useful for the next discussions.  Point S represents a tiny sound source, called a point source. It emits sound waves in all directions.  Wavefronts are surfaces over which the oscillation due to the sound wave have the same value.  Rays are directed lines perpendicular to the wavefronts that indicate the direction of travel of the wavefronts.

4/6 January–April 2010University Physics: Wave and Electricity Sound Waves  As a longitudinal wave, sound wave travels through a medium (solid, liquid, or gas), involving oscillations parallel to the direction of wave travel.  When a sound wave moves in time, the displacement of air molecules, the pressure, and the density vary sinusoidally with the frequency of the vibrating source.

4/7 January–April 2010University Physics: Wave and Electricity Interference  Like transverse waves, sound waves can undergo interference.  Now we will consider, in particular, the interference between two identical sound waves traveling in the same direction.  Two point sources S 1 and S 2 emit sound waves that are in phase and of identical wavelength λ.  Thus, as the waves emerge from the sources, their displacements are always identical.  The waves travels through point P, with the distance L 1 or L 2 much greater than the distance between the sources, S 1 and S 2.  The two sources can be approximated to travel in the same direction at P.

4/8 January–April 2010University Physics: Wave and Electricity Interference  From the figure, the path L 2 traveled by the wave from S 2 is longer than the path L 1 traveled by the wave from S 1.  The difference in path lengths means that the waves may not be in phase at point P.  The phase difference Φ at P depends on their path length difference,  The relation between phase difference to path length difference, as we recall from previous chapter, is:

4/9 January–April 2010University Physics: Wave and Electricity  Fully constructive interference occurs when Φ is zero, 2π, or any integer multiple of 2π. Fully constructive interference  Fully destructive interference occur when Φ is an odd multiple of π, Interference Fully destructive interference

4/10 January–April 2010University Physics: Wave and Electricity Interference Fully destructive, arrive “out of phase” Fully constructive, arrive “in phase”

4/11 January–April 2010University Physics: Wave and Electricity Example: Interference Two point sources S 1 and S 2, which are in phase and separated by distance D = 1.5λ, emit identical sound waves of wavelength λ. (a)What is the path length difference of the waves from S 1 and S 2 at point P 1, which lies on the perpendicular bisector of distance D, at a distance greater that D from the sources? What type of interference occurs at P 1 ? The waves undergo fully constructive interference at P 1

4/12 January–April 2010University Physics: Wave and Electricity Example: Interference (b)What are the path length difference and type of interference at point P2?P2? The waves undergo fully destructive interference at P 2

4/13 January–April 2010University Physics: Wave and Electricity Example: Interference (c) (c) The figure below shows a circle with a radius much greater than D, centered on the midpoint between sources S1 S1 and S 2. What is the number of points N around this circle at which the intereference is fully constructive? Using the symmetry, as we go around the circle, we will find 6 points where the interference is a fully constructive interference

4/14 January–April 2010University Physics: Wave and Electricity Homework 4: Two Speakers Two speakers separated by distance d 1 = 2 m are in phase. A listener observes at distance d 2 = 3.75 m directly in front of one speaker. Consider the full audible range for normal human hearing, 20 Hz to 20 kHz. Sound velocity is 343 m/s. (a)What is the lowest frequency f min,1 that gives minimum signal (destructive interference) at the listener’s ear? (b)What is the second lowest frequency f min,2 that gives minimum signal? (c)What is the lowest frequency f max,1 that gives maximum signal (constructive interference) at the listener’s ear? (d)What is the highest frequency f max,n that gives maximum signal?

4/15 January–April 2010University Physics: Wave and Electricity Homework 4: Other Two Speakers New Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 12 m to the right of speaker A. The frequency of the waves emitted by each speaker is 686 Hz. Sound velocity is 343 m/s. You are standing between the speakers, along the line connecting them, and are at a point of constructive interference. (a)How far must you walk toward speaker B to move to a point of destructive interference? (b)How far must you walk toward speaker B to move to another point of constructive interference?