Chapter 15 Mechanical Waves © 2016 Pearson Education, Inc.

Learning Goals for Chapter 15 Understand relationship between speed, frequency, & wavelength for a periodic wave. Calculate speed of waves on a rope or string. Identify when mechanical waves overlap & interfere. Understand properties of standing waves on a string Grasp how stringed instruments produce sounds of specific frequencies.

Introduction Earthquake are waves that carry enormous power Other types of mechanical waves, e.g. sound waves or the vibration of the strings of a piano, also carry (far less) energy. Overlapping waves “interfere” which helps us understand musical instruments.

Types of mechanical waves Wave on string is a mechanical wave. Hand moves string up & down, producing transverse wave. Wave moves to the right with a speed.

Types of mechanical waves Pressure waves in a fluid are also mechanical waves. Piston moves right, compressing gas or liquid, & returns, producing a longitudinal wave Wave moves to the right with a speed. Particles move right and left

Types of mechanical waves Surface wave on liquids are mechanical waves. Board moves right & then returns, producing a combination of longitudinal & transverse waves.

Mechanical waves “Doing the wave” at a sports stadium is a mechanical wave. Wave disturbance propagates through crowd, but there is no transport of matter. None of the spectators moves from one seat to another.

Periodic waves For periodic waves, each particle of medium undergoes periodic motion. Wavelength λ (“lambda”) of periodic wave is length of one complete wave pattern (units: Distance!) Speed of periodic wave of frequency f is:

Periodic transverse waves Mass attached to spring undergoes SHM, producing sinusoidal wave traveling right on string: Wavelenth l

Periodic transverse waves Mass attached to spring undergoes SHM, producing sinusoidal wave traveling right on string:

Periodic waves Series of drops falling into water produces a periodic wave that spreads radially outward. Wave crests & troughs are concentric circles. Wavelength λ is radial distance between adjacent crests or adjacent troughs.

Periodic longitudinal waves Consider long tube filled with fluid, with piston at left end. Push piston in, compress fluid near piston This region pushes against neighboring region of fluid, And so on… and a wave pulse moves along the tube.

Periodic longitudinal waves

Mathematical description of a wave Wave function for sinusoidal wave moving in +x-direction: y = displacement of particle at time t & position x (distance) A = amplitude of wave (distance) k = wave number, defined as k = 2π/λ (m-1) ω = angular frequency, defined as ω = 2πf = 2π/T (rad/sec)

Graphing the wave function y(x, t) =𝐴𝑐𝑜𝑠 𝑘𝑥−𝑤𝑡 𝑦 𝑥 = 𝐴𝑐𝑜𝑠 𝑘𝑥 @ 𝑡=0

Graphing the wave function y(x, t) at a particular time t is a SNAPSHOT in time graph of the entire wave at that instant 𝑦 𝑥 = 𝐴𝑐𝑜𝑠 𝑘𝑥 @ 𝑡=0

Graphing the wave function y(x, t) =𝐴𝑐𝑜𝑠 𝑘𝑥−𝑤𝑡 𝐴𝑐𝑜𝑠 𝑤𝑡 @ 𝑥=0

Graphing the wave function y(x, t) =𝐴𝑐𝑜𝑠 𝑘𝑥−𝑤𝑡 y(x, t) at a particular position x is a MOVIE graph of the wave at that point only 𝐴𝑐𝑜𝑠 𝑤𝑡 @ 𝑥=0

Particle velocity and acceleration in a sinusoidal wave

Particle velocity and acceleration in a sinusoidal wave

The speed of a wave on a string Key property of any wave is wave speed. Consider string with tension F & linear mass density (mass per unit length) is μ. Expect speed of transverse waves on string v to increase when tension F increases Expect wave speed to decrease when μ increases. Wave speed is:

The speed of a wave on a string Transmission cables w/ relatively large mass per unit length, & low tension. If cables disturbed—say, by birds —transverse waves will travel at a slow speed.

Power in a wave Instantaneous power in a sinusoidal wave Power is never negative! Energy never flows opposite to the direction of wave propagation.

Power in a wave Waves transfer power because they transfers energy. Average power proportional to square of amplitude Average power proportional to square of frequency. This result is true for all waves. For transverse wave on string, average power is:

Wave intensity Intensity of a wave is average power it carries per unit area. Units: Watts/m2 IF waves spread out uniformly in all directions & NO energy is absorbed I at any distance r from a wave source is inversely proportional to r2.

Reflections of wave pulses @ fixed end What happens when a wave pulse or a sinusoidal wave arrives at the end of the string? If end is fastened to a rigid support, it is a fixed end that cannot move. Arriving wave exerts force on fixed support….

Reflection of a wave pulse at a fixed end of a string Reaction force exerted by support on string “kicks back” on string Reaction force sets up a reflected pulse or inverted wave traveling in reverse direction.

Reflection of wave pulse at free end Free end is one that is perfectly free to move in direction perpendicular to length of string. When wave arrives at free end, ring slides along rod, reaching maximum displacement, coming momentarily to rest…

Reflection of a wave pulse at a free end of a string String is now stretched, giving increased tension, Free end is pulled back down A reflected pulse is produced But NOT inverted!

Superposition Interference is result of overlapping waves at a point in time. Principle of superposition: When two or more waves overlap, total displacement is sum of displacements of individual waves.

Superposition Overlap of two wave pulses— one right side up, one inverted—traveling in opposite directions. Time increases from top to bottom.

Superposition Overlap of two wave pulses— one right side up, one inverted—traveling in opposite directions. Time increases from top to bottom.

Standing waves on a string Waves traveling in opposite directions on a taut string interfere with each other. Result is standing wave pattern that does not move on the string. Destructive interference occurs where the wave displacements cancel, and constructive interference occurs where the displacements add. At the nodes no motion occurs, and at the antinodes the amplitude of the motion is greatest.

Standing waves on a string Waves traveling in opposite directions on a taut string interfere with each other. Result is standing wave pattern that does not move on the string.

Standing waves on a string This pattern is called the second harmonic.

Standing waves on a string As frequency of oscillation of right-hand end increases, pattern of standing wave changes. More nodes & antinodes are present in higher frequency standing wave.

The mathematics of standing waves Derive wave function for standing wave by adding two wave functions for waves w/ equal amplitude, period, & wavelength traveling in opposite directions. Wave function for standing wave on string where x = 0 is a fixed end is: Standing-wave amplitude ASW is twice the amplitude A of either of the original traveling waves: ASW = 2A.

Normal modes f1 is the fundamental frequency Wavelength l1 = 2 L Wave speed v = f1 l1

Normal modes f1 is the fundamental frequency f2 is the second harmonic (first overtone) Wavelength l2 = L Wave speed v = f2 l2 Wave speed v hasn’t changed! Same T, same m!

Normal modes f1 is the fundamental frequency, f2 is the second harmonic (first overtone), f3 is the third harmonic (second overtone), etc.

Normal modes For a taut string fixed at both ends, possible wavelength: Possible frequencies are fn = n v/2L = nf1 where n = 1, 2, 3, …

Standing waves and string instruments Move string on a musical instrument (pluck, bow, or strike), a standing wave with fundamental frequency is produced: This is also frequency of sound wave created in surrounding air by vibrating string. Increasing tension F increases frequency (and pitch).