General Physics waves_2 Group Work Add these two waves together. Demo equipment: coil spring and table clamp; wave tank, bucket, block, plunger, cork floats; corrugated resonance tube

General Physics waves_2 Part 2: resonance and more dimensions

General Physics waves_2 Objectives Evaluate combinations of waves from confined systems. Explain the refraction of waves. Explain the behavior of ocean waves near the shore.

General Physics waves_2 Combinations of Waves

General Physics waves_2 Standing Waves Sum of waves of equal amplitude and wavelength traveling in opposite directions Half-wavelength divides exactly into the available space Wave pattern has locations of minimum and maximum variation (nodes and antinodes) (standing longitudinal waves) Run at: w1 = 0.2; k1 = 0.2; ampl = 15 integral multiples (half, third, quarter) of lambda: multiply w1, k1 by 2, 3, 4 add same-lambda wave with negative amplitude Beats: slightly vary w2 and k2 together from wave 1 values (0.22 and 0.22; 0.21 and 0.21, etc.) Standing waves (use w of about 0.2; try w = 0.2, k = 0.1)

General Physics waves_2 Resonance Objects have characteristic frequencies at which standing waves are sustained Lowest frequency = fundamental Higher frequencies = overtones Sustained motion is a combination of normal modes

Vibrational Modes: Clamped String General Physics waves_2 Vibrational Modes: Clamped String Insert Figure 15.3 from class text Source: Griffith, The Physics of Everyday Phenomena, Figure 15.13

General Physics waves_2 Poll Question Bowing a ‘cello string 1/4 of the way along will excite which harmonic the MOST? The fundamental. The first overtone. The second overtone. The third overtone. Most energy into antinode of first overtone.

Group Work Add together: a fundamental and second overtone.

General Physics waves_2 Group Work Result Add together: a fundamental and first overtone. Approaching a sawtooth wave

General Physics waves_2 Group Work Result Add together: a fundamental and second overtone. Approaching a square wave

Combinations of Harmonics General Physics waves_2 Combinations of Harmonics Characteristic sounds arise from combining particular harmonics in specific ratios Fourier analysis suimulation flute oboe saxophone

“Closed” and “Open” Tube Modes General Physics waves_2 “Closed” and “Open” Tube Modes Source: Halliday, Resnick, and Walker, Fundamentals of Physics, 2003, p 419.

General Physics waves_2 Poll Question Which has the lowest fundamental tone? A closed tube of length L. An open tube of length L. Both have the same fundamental tone. Not enough information.

General Physics waves_2 Sequence of Harmonics Western musical scale and harmonies are based on overtone series (sound files) Sound files: overtones of open tube or clamped string

Musical Tones Octave higher = 2 the frequency Octave + fifth: 3 the frequency Even-tempered scale = compromise to facilitate transposition 12 (half-) steps per octave Half-step: 2  the frequency 12

Two-Dimensional Waves General Physics waves_2 Two-Dimensional Waves Ocean waves Earthquake surface waves Animation rectangular membrane Animation circular membrane

Wave Refraction Direction changes to keep wave front continuous Source: University of Southampton, Institute for Sound and Vibration Research

Wave Refraction Snell’s Law: u2 sinq1 = u1 sinq2 q1 q2 Speed u1

Question Waves approach a slow zone at an angle. The direction the waves travel does not change. bends toward the interface. bends away from the interface. C slow A B fast

Refraction Example u = 1/2 u = 3/4 u = 1

Water Waves Restoring force is gravity Fastest in deep water, slow in shallow water Breaking and refractive behavior in shallow water Speed depends on wavelength Very complicated behavior!

Question Ocean waves approaching a shore will bend so that the wave fronts make a greater angle with the shore. bend so that the wave fronts are more parallel to the shore. veer away from the shore. not change their direction.

General Physics waves_2 Ocean Wave Refraction

Group Work Sketch how the train of waves would move into the region of slow speed. fast slow

General Physics waves_2 Breaking Wave

Shoaling and Breaking Waves slow in shallows by shore Wavelength shortens and amplitude rises Troughs move slower than crests Crests fall forward

Question When waves approach the shore, their increases and decreases. speed, mass force, mass wavelength, period amplitude, frequency amplitude, wavelength frequency, wavelength

Tsunami