Chapter 11 Vibrations and Waves
Driven Harmonic Motion and Resonance RESONANCE Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. Resonance occurs when the frequency of the force matches a natural frequency, f 0, at which the object will oscillate. f 0 = 1 2π k m
The Nature of Waves 1. A wave is a traveling disturbance. 2. A wave carries energy from place to place.
The Nature of Waves Longitudinal Wave - the disturbance caused by the wave moves along the direction that the wave propagates, e.g., sound waves, compressed slinky waves
The Nature of Waves Transverse Wave - the disturbance caused by the wave moves perpendicular to the direction that the wave propagates, e.g., water waves, shaken slinky waves, electromagnetic waves.
The Nature of Waves Water waves are partially transverse and partially longitudinal.
Periodic Waves Periodic waves consist of cycles or patterns that are produced over and over again by the source. In the figures, every segment of the slinky vibrates in simple harmonic motion, provided the end of the slinky is moved in simple harmonic motion.
Periodic Waves In the drawing, one cycle is shaded in color. The amplitude A is the maximum excursion of a particle of the medium from the particles undisturbed position. The wavelength is the horizontal length of one cycle of the wave. The period is the time required for one complete cycle. The frequency is related to the period and has units of Hz, or s -1. f = 1 T
Periodic Waves The propagation velocity of a periodic wave is related to its frequency and wavelength. Consider the motion of a long train as a periodic wave which repeats itself with the passing of each identical car: Since velocity is distance/time è λ v = = T fλ
Periodic Waves Example: The Wavelengths of Radio Waves AM and FM radio waves are transverse waves consisting of electric and magnetic field disturbances traveling at a speed of 3.00x10 8 m/s. A station broadcasts AM radio waves whose frequency is 1230x10 3 Hz and an FM radio wave whose frequency is 91.9x10 6 Hz. Find the distance between adjacent crests in each wave. λ v = = T fλ λ = v f
Periodic Waves AM λ = f v = 8 3.00 10 m s 3 1230 10 Hz = 244 m FM λ = f v = 3.00 10 91.9 10 8 6 m s Hz = 3.26 m
The Speed of a Wave on a String The speed at which the wave moves to the right depends on how quickly one particle of the string is accelerated upward in response to the net pulling force. v = F m L tension linear density
The Speed of a Wave on a String Example: Waves Traveling on Guitar Strings Transverse waves travel on each string of an electric guitar after the string is plucked. The length of each string between its two fixed ends is 0.628 m, and the mass is 0.208 g for the highest pitched E string and 3.32 g for the lowest pitched E string. Each string is under a tension of 226 N. Find the speeds of the waves on the two strings.
The Speed of a Wave on a String High E v = F m L = 226 N ( 3 0.208 10 kg) ( 0.628 m) - = 826m s Low E v = F m L = 226 N ( 3 3.32 10 kg) ( 0.628 m) - = 207m s
The Speed of a Wave on a String Conceptual Example: Wave Speed Versus Particle Speed Is the speed of a transverse wave on a string the same as the speed at which a particle on the string moves?
Speed of longitudinal waves LIQUIDS SOLID BARS v = B ρ v = Y ρ B bulk modulus Y Young's modulus
Example: Compare the speed of a wave in water to a wave in a bar of aluminum. Water: B = 2.2 10 9 N/m 2 ρ =1000 kg/m 3 v = B 2.2 109 = ρ 1000 =1500 m/s Aluminum: Y = 6.9 10 10 N/m 2 ρ = 2700 kg/m 3 v = Y 6.9 1010 = ρ 2700 = 5100 m/s
Intensity Waves carry energy that can be used to do work. The amount of energy transported per second is called the power of the wave. The intensity is defined as the power that passes perpendicularly through a surface divided by the area of that surface. I = P A
Intensity Example: Sound Intensities 12x10-5 W of sound power passed through the surfaces labeled 1 and 2. The areas of these surfaces are 4.0 m 2 and 12 m 2. Determine the sound intensity at each surface.
Intensity I 12 10 W 5 5 2 1 = = 3.0 10 W m 2 1 4.0m = A P I 12 10 W 5 5 2 2 = = 1.0 10 W m 2 2 12m = A P
Intensity If the source emits the wave uniformly in all directions, the intensity depends on the distance from the source in a simple way. power of wave source I = P 4π r 2 area of sphere
Intensity I = P 4π r 2 I 1 r 2 I 1 = r 2 2 2 I 2 r 1 For SHM E = 1 2 ka2 I P E A 2 I 1 = r 2 2 I 2 r = A 2 1 r 2 = A 1 2 2 1 A 2 r 1 A 2
Example: The intensity of an earthquake wave passing through the Earth is measured to be 2.5 10 6 J/(m 2 s) at a distance of 44 km from the source. What was its intensity when it passed a point only 1.0 km from the source? At what rate did energy pass through an area of 7.0 m 2 at 1.0 km? I 1 I 2 = r 2 2 r 1 2 I 1 = I 2 r 2 2 ( ) 442 r 1 2 = 2.5 106 1 2 = 4.8 10 9 J/(m 2 s) P = I 1 A 1 = ( 4.8 10 9 )( 7.0) = 3.4 10 10 J/s
Wave Fronts and Rays Defining wave fronts and rays. Consider a sound wave since it is easier to visualize. Shown is a hemispherical view of a sound wave emitted by a pulsating sphere. The rays are perpendicular to the wave fronts (e.g. crests) which are separated from each other by the wavelength of the wave, λ.
Wave Fronts and Rays The positions of two spherical wave fronts are shown in (a) with their diverging rays. At large distances from the source, the wave fronts become less and less curved and approach the limiting case of a plane wave shown in (b). A plane wave has flat wave fronts and rays parallel to each other.
Reflection of string pulses at boundaries and interfaces Reflection inverted Reflection not inverted } } From less dense medium to more dense medium From more dense medium to less dense medium
Law of reflection of waves at a boundary or interface: θ i =θ r