Chapter 6 Mechanical Waves A wave is a disturbance that travels, or propagates, without the transport of matter. Examples: sound/ultrasonic wave, EM waves, and earthquake wave. Mechanical waves, such as water waves or sound waves, travel within, or on the surface of, a material with elastic properties. Electromagnetic waves, such as radio, microwave, and light, can propagate through a vacuum. Matter waves discovered in elementar particles, can displa wavelike behavior. 6. Wave Characteristics In a transverse wave, shown in Fig. (a), the displacement of a particles is perpendicular to the direction of travel of the wave. In a longitudinal wave, shown in Fig. (b), the displacement of the particles is along the direction of wave propagation. A solid can sustain both kinds of waves, however, an ideal fluid (nonviscous) having no well-defined form can onl propagate longitudinal waves. Seawater, one kind of fluid, can sustain both transverse and longitudinal waves. Wh?
Motion of the Medium The particles of the medium are not carried along with the wave. The undergo small displacements about an equilibrium position, whereas the wave itself can travel a great distance. Since the particles of the medium does not travel with the wave, what does? Energ and momentum. 3 6. Superposition of Waves Principle of linear superposition: The total wave function at an point is the linear sum of the individual wave function; that is, + + + L+ T 3 N In all our examples we assume that linear superposition is valid. What is the interference? constructive or destructive. 4
6.3 Speed of a pulse on a string The speed at which a pulse propagates depends on the properties of the medium. In the pulse frame from Newton s second law we have F sinθ mv R The mass in segment AB is mµr, where µ is the linear mass densit. In small angle limit, we have v µ F restoring force factor Inertia factor 5 Example 6. One end of a string is fixed. It has over a pulle and has a block of mass.0 kg attached to the other end. The horizontal part has a length of.6 m and a mass of 0 g. What is the speed of a transverse pulse on the string. Solution: Tension is simpl the weight of the block; that is F9.6 N. The linear mass densit is µ0.05 kg/m. The speed is v µ F 9.6 0.05 39.6 m/s How guitar works? Same concept. 6
6.4 Reflection and Transmission The reflection of a pulse for different end conditions, fixed or free. What is the boundar condition of each case? 7 Difference Perspective: Superposition of the actual pulse and an imaginar pulse 8
Between the Extreme Cases of a Fixed End or a Free End from light string to heav string from heav string to light string This results in partial reflection and transmission. Since the tensions are the same, the relative magnitudes of the wave velocities are determined b mass densities. 9 6.5 Traveling Waves In the stationar frame, the pulse has same shape but is moving at a velocit v. In the moving frame the pulse is at rest, so the vertical displacement at position x is given b some function f(x ) that describe the shape of the pulse. ( x ) f ( x ) x x vt ( x, f ( x v ( x, f ( x ± v Plus or minus sign represents forward or backward propagation. 0
6.6 Traveling Harmonic Waves If the source of the waves is a simple harmonic oscillator, the function f(x±v is sinusoidal and it represents a traveling harmonic wave. The stud of harmonic wave is particularl important because a disturbance of an shape ma be formed b adding together suitable harmonic components of different frequencies and amplitudes. Traveling Harmonic Waves (II) ( x, Asin( kx ±ω t + φ) A:amplitude k π λ : wave number ω π T v λ T ω k T : period λ : wavelength f :frequenc : angular frequenc λf : velocit
The equation of a wave is Example 6.3: π x, 0.05sin[ (0x 40 π ]m ( 4 Find: (a) the wavelength, the frequenc, and the wave velocit; (b) the particle velocit and acceleration at x0.5 m and t0.05 s. Solution: (a) A0.05, k5π, ω0π, φ-π/4, λ/5, f0, and T0. (b) d dt d dt π cos π π [ (0x 40 ] 0π sin π π [ (0x 40 ] 40 m/s 4. m/s 4 3 6.7 Standing Waves Two harmonic waves of equal frequenc and amplitude traveling through a medium in opposite directions form a standing wave pattern. ( x, ( x, + Asin( kx + ωt + φ) Asin( kx ωt + φ) Asin( kx + φ) cos( ω Nodes: the points of permanent zero displacement. Antinodes: the points of maximum displacement. 4
6.8 Resonant Standing Waves on a String How to generate an opposite propagating wave with equal amplitude and frequenc? Reflection. Wh the resonant frequenc is discrete? Boundar condition. 5 Resonant Standing Waves (II) What is the minimum frequenc allowable on the string? If the propagating velocit is the same, then the minimum frequenc corresponds to the longest wavelength. To satisf the boundar condition, the maximum wavelength is λl. Fundamental frequenc (first harmonic) λ L f v λ v L 6
Resonant Standing Waves: (III) Normal modes Normal modes λ f n n L n nv L 7 Resonant Standing Waves: (IV) Resonant frequencies A simple wa to set up a resonant standing is shown below. One prong of a tuning fork is attached to one end of the string. The string hangs over a pulle and a weight determine the tension in it. n n f n v L L F µ Example: guitar Question: Which mode will dominate? 8
Resonant Standing Waves: (V) D normal modes 9 6.9 The Wave Equation One can obtain the wave equation b taking the partial derivatives of the wave function for the traveling harmonic wave. with respect to t and x: ( x, sin( kx ω t + φ) d dx d dt k o o ω o sin( kx ωt + φ) sin( kx ωt + φ) B comparing these derivatives, we see that d d dx v dt 0
6.0 Energ Transport on a String Linear energ densit (J/m): de dx dk dx + du dx µ ( ω ) cos ( kx ω o Average power (J/s): P av de dt de dx dx dt µ ( ω o ) v 6. Velocit of Waves on a String From Newton s second law we have F[sin( θ + θ ) sin( θ )] ( µ x) t Small angle approximation: sinθ tanθ x ( µ / F) x t
6.3 Speed of a pulse on a string The speed at which a pulse propagates depends on the properties of the medium. In the pulse frame from Newton s second law we have F sinθ mv R The mass in segment AB is mµr, where µ is the linear mass densit. In small angle limit, we have v µ F restoring force factor Inertia factor 3 Exercises and Problems Ch.6: Ex. 0,, 9,, 35, 39 Prob. 6, 9, 0, 4