Oscillations (continues) Vibrating systems What we studied before? Simple harmonic motion (one mode) Mass on spring Simple pendulum (SHM for small angles) More complicated systems More than one mode of oscillations Oscillations that are not harmonic 1

1. Two-mass vibrator m m m m Longitudinal vibrations Modes (a) and (b) are independent Transverse vibrations mm m m Two-mass vibrator has 2 longitudinal and 4 transverse modes of vibration (Violin string normally vibrates transversely) 2

3 2. Systems with many modes N-mass vibrator has N longitudinal and 2N transverse modes of vibration

3. Standing waves on a string Vibrating string can be thought of as the limit of the mass-spring system when the number of masses becomes very large 4

Standing waves on a string n=1,2,3... 5

4. Vibrations in musical instruments 1.Strings 2.Membranes 3.Bars 4.Plates (Chladni patterns) 5.Tuning forks 5. Vibration Spectra (Fourier Analysis) 6

1. There are many examples of waves: - Seismic waves - Ripples on a pond - Sound - Electromagnetic wave including light Waves A wave can have a short duration – just a quick pulse; these are called “wave pulses” A wave can also repeat; these are called continuous waves 2. Types of Waves A transverse wave is a wave in which the oscillations are transverse to the direction the wave travels A longitudinal wave is a wave in which the oscillations are along the direction the wave travels 7

3. Wavelength and Speed of Waves “Speed” of a wave is the rate of movement of the disturbance. It is not the speed of the individual particles! Speed is determined by the properties of the medium. λ - wave length T - period f - frequency t y T x yλ Sinusoidal waves Angular frequency: Wave number: 8

Question 2: A sinusoidal wave has a wavelength of 4.00 m and an angular frequency of 3.14 rad/s. What is the speed of the wave? Question 3: If you double the wavelength of a periodic wave and the wave speed does not change, what happens to the wave frequency f ? 1) f is unchanged2) f is halved3) f is doubled Question 1: A sinusoidal wave has a wavelength of 4.00 m and a period of 2.00s. What is the speed of the wave? 9

Example: A metal string has a mass of kg, a length of 40 cm, and is under 80 N tension. 1. What is the speed of a wave in this metal string? 2. What happed if we increase a tension up to 120 N? 4. Speed of waves in strings The speed of the wave increases if we increase the tension (F) in the string, and decreases if we increase the linear mass density (μ) of the string. 10

Question 1: A wave pulse can be sent down a rope by jerking sharply on the free end. If the tension of the rope is increased, how will that affect the speed of the wave? 1) speed increases2) speed does not change3) speed decreases Question 2: A wave pulse is sent down a rope of a certain thickness and a certain tension. A second rope made of the same material is twice as thick, but is held at the same tension. How will the wave speed in the second rope compare to that of the first? 1) speed increases2) speed does not change3) speed decreases Thicker rope means more massive rope Question 3: A length of rope L and mass M hangs from a ceiling. If the bottom of the rope is jerked sharply, a wave pulse will travel up the rope. As the wave travels upward, what happens to its speed? Keep in mind that the rope is not massless. 1) speed increases2) speed does not change3) speed decreases The tension in the rope is not constant in the case of a massive rope! The tension increases as you move up higher along the rope, because that part of the rope has to support all of the mass below it! Since the tension increases as you go up, so does the wave speed. 11