Mechanical Waves

Mechanical Waves Ch 21-23

Waves • A wave is a disturbance in a medium • which carries energy from one point to another • without the transport of matter. • The medium allows the disturbance to propagate. Physics chapters 21-23

Transverse Wave • Particles oscillate at right angles to the direction of motion. Physics chapters 21-23

Longitudinal Waves • Particles oscillate parallel to the direction of motion. Physics chapters 21-23

Periodic Waves & Pulses • A wave pulse is a single disturbance. • A periodic wave is a series of disturbances or wave train. Physics chapters 21-23

Transverse Wave Speed • Determined by the medium and its properties. • elasticity or restoring force • inertia Physics chapters 21-23

Wave on a mediumwith tension. • String, rope, wire, etc… • T is the tension, & m is the linear density, m = m/L = mass per unit length. Physics chapters 21-23

Waves • Speed: Physics chapters 21-23

Wave Terminology • Frequency (f) - cycles per second. (Hz) • Period (T) - Seconds per cycle. • Amplitude (A) - Maximum displacement from equilibrium. • The distance that a wave travels in one period is the wavelength (l). Physics chapters 21-23

Example 1 • A wave travels along a string. The time for a particular point to move from a maximum displacement to zero is 0.170 s. The wavelength is 1.40 m. What are the period, frequency, and wave speed? Physics chapters 21-23

Example 1 continued • It takes 0.680 s for one cycle, so T = 0.680 s • f = 1/T, so f = 1.47 Hz Physics chapters 21-23

Example 2 • What is the speed of a transverse wave in a rope of length 2.00 m and mass 60.0 g under a tension of 500 N? Physics chapters 21-23

Example 2 continued Physics chapters 21-23

Polarization • Most transverse waves are linearly polarized • They either move just up and down • Vertically polarized • Or just side to side • Horizontally polarized Physics chapters 21-23

Circular polarization • If we combine two perpendicular waves that have equal amplitude but are out of step by a quarter-cycle, the resulting wave is circularly polarized. Physics chapters 21-23

Polarizing filters • Only let through waves that are polarized one way. • Like passing a rope through a slot in a board – only waves in the direction of the slot will get through. Physics chapters 21-23

Longitudinal Wave Speed • Depends on the pressure change and the fractional volume change • Where r is the density. B is the bulk modulus of a fluid. Y is young’s modulus for a solid. See tables 12-1 and 12-2. B = 1/k Physics chapters 21-23

Longitudinal waves • Don’t have polarization • When the frequency is within the range of human hearing, it is called sound. Physics chapters 21-23

Sound waves in gases • Temperature doesn’t remain constant as sound waves move through air. • So, we use the equation • Where g is the ratio of heat capacities (ch 18), R is the ideal gas constant (8.314 J/mol∙K), T is temperature in K, and M is the molecular mass (ch 17). Physics chapters 21-23

Sound waves • Humans can hear from about 20 Hz to about 20 000 Hz. • Air is not continuous – it consists of molecules. • Like a swarm of bees. • Also sort of like wave/particle duality. Physics chapters 21-23

Mathematical wave description y(x, t) = A sin(wt – kx) (Motion to right) or y(x, t) = A sin(wt + kx) (Motion to left) Physics chapters 21-23

Reflection • When a wave comes to a boundary, it is reflected. • Imagine a string that is tied to a wall at one end. • If we send a single wave pulse down the string, • when it reaches the wall, it exerts an upward force on the wall. Physics chapters 21-23

Reflection • By Newton’s third law, • the wall exerts a downward force that is equal in magnitude. • This force generates a pulse at the wall, which travels back along the string in the opposite direction. Physics chapters 21-23

Reflection • In a ‘hard’ reflection like this, • there must be a node at the wall • because the string is tied there. • The reflected pulse is inverted from the incident wave. Physics chapters 21-23

Reflection • Now imagine that instead of being tied to a wall • the string is fastened to a ring which is free to move along a rod. • When the wave pulse arrives at the rod, the ring moves up the rod • and pulls on the string. Physics chapters 21-23

Reflection • This sort of ‘soft’ reflection • creates a reflected pulse • that is not inverted. Physics chapters 21-23

Transmission • When a wave is incident on a boundary that separates two regions of different wave speeds • part of the wave is reflected • and part is transmitted. Physics chapters 21-23

Transmission • If the second medium is denser than the first • the reflected wave is inverted. • If the second medium is less dense • the reflected wave is not inverted. • In either case, the transmitted wave is not inverted. Physics chapters 21-23

Transmission Physics chapters 21-23

Interference Physics chapters 21-23

Interference • The effect that waves have when they occupy the same part of the medium. • They can add together or cancel each other out. • After the waves pass each other, they continue on with no residual effects. Physics chapters 21-23

Constructive Interference Physics chapters 21-23

Constructive Interference • l out of phase = 360° = 1 cycle = 2p rad Physics chapters 21-23

Destructive Interference Physics chapters 21-23

Destructive Interference • l/2 out of phase = 180° = 1/2 cycle = p rad Physics chapters 21-23

Superposition of waves • If two waves travel simultaneously along the same string • the displacement of the string when the waves overlap • is the algebraic sum of the displacements from each individual wave. Physics chapters 21-23

Standing Waves • Consider a string that is stretched between two clamps, like a guitar string. • If we send a continuous sinusoidal wave of a certain frequency along the string to the right • When the wave reaches the right end, it will reflect and travel back to the left. Physics chapters 21-23

Standing waves • The left-going wave the overlaps with the wave that is still traveling to the right. • When the left-going wave reaches the left end • it reflects again and overlaps both the original right-going wave and the reflected left-going wave. • Very soon, we have many overlapping waves which interfere with each other. Physics chapters 21-23

Standing waves • For certain frequencies • the interference produces a standing wave pattern • with nodes and large antinodes. • This is called resonance • and those certain frequencies are called resonant frequencies. Physics chapters 21-23

Standing waves • A standing wave looks like a stationary vibration pattern, • but is the result of waves moving back and forth on a medium. Physics chapters 21-23

Standing waves • Superposition of reflected waves which have a maximum amplitude and appear to be a stationary vibration pattern. y1 + y2 = -2Acos(wt)sin(kx) Physics chapters 21-23

Standing Waves • If the string is fixed at both ends • there must be nodes there. • The simplest pattern of resonance that can occur is one antinode at the center of the string. Physics chapters 21-23

Standing Waves on Strings • Nodes form at a fixed or closed end. • Antinodes form at a free or open end. Physics chapters 21-23

Standing waves • For this pattern, half a wavelength spans the distance L. • This is called the 1st harmonic. • It is also called the fundamental mode of vibration. Physics chapters 21-23

Standing waves • For the next possible pattern, a whole wavelength spans the distance L. • This is called the 2nd harmonic, or the 1st overtone. Physics chapters 21-23

Standing Waves • For the next possible pattern, one and a half wavelengths span the distance L. • This is called the 3rd harmonic, or the 2nd overtone. Physics chapters 21-23

Standing waves • In general, we can write Physics chapters 21-23

Standing Waves Physics chapters 21-23

Standing Waves on a String Physics chapters 21-23

Mechanical Waves