Long spring A wave travels down a long spring at a speed of 20 m/s. It is generated by a hand moving up and down 10 times per second, with a total displacement of 6 cm. At time zero, the hand is at the lowest point. Find an equation describing the vertical displacement of the spring as a function of position and time.

Speed of wave on a string Elastic forces Mass per length of string A piano string is under a tension of about 2000 N, with a mass of 2.5 g per meter of length. Determine the speed of the transverse wave on this string.

Speed of wave on a string Consider a long string consisting of a thin and a thick segment. Which of the three quantities will change, if …the wave passes from the thin to the thick segment? …the frequency of the source changes?

Reflection at a fixed end Reflected pulse is inverted

Reflection at a loose end Reflected pulse is not inverted

Superposition of waves

Constructive and destructive superposition

Beats

Standing waves on strings A string of length L, fixed at both ends, has resonance frequencies: Fundamental 1 st Harmonic 1 st overtone 2 nd Harmonic 2 nd overtone 3 rd Harmonic Eigen frequencies of the string:

Length of the piano string The speed of transverse waves on a piano string is 942 m/s. How long does this string have to be in order to generate a sound Hz (middle C)?

Resonances of tubes Fundamental 1 st overtone Closed tubes open tubes

Sound and hollow tubes Sound: transverse wave, compression wave node Loop at open end node Loop at open end Both ends open: One end closed:

Sound: speed, Doppler effect

Sound wave density rarefaction densification pressure expansion compression Fluctuations: appr. 0.01%

Speed of sound Compression waves Steel: Strong forces, high density

Speed of sound Compression waves Water Medium forces, small distances

Speed of sound Compression waves Air Nearly no forces, large distances Temperature dependent!

Classification Growing frequency 20 Hertz Hertz=20kHz Audible range 3000 Hz best hearing Infra sound Ultra sound Growing Wave length? Earthquakes, heavy traffic Medical imaging, dog whistles, bats

Wave lengths of the audible range 20 kHz: 20 Hz: Width of a finger Width of a house Objects in our environment are of the same order of size as the wave length of sound waves.

Sound and hollow tubes Sound: transverse wave, compression wave node Loop at open end node Loop at open end Both ends open: One end closed:

Resonances of tubes Fundamental 1 st overtone Closed tubes open tubes

Human ear The human ear canal is a tube of approximately 3 cm length. Determine the fundamental frequency.

Doppler Effect

Resting sound source source at rest observer at rest Frequency f s Frequency f o V=340m/s

Sound source moving toward observer sourceobserver at rest Frequency f s Frequency f o Observer hears increased pitch (shorter wave length)

Sound source moving away from observer source observer at rest Frequency f s Frequency f o Observer hears decreased pitch (longer wave length)

Frequency heard by observer Source moving toward observer. Perceived wavelength: With Observer:

Observed frequency Observer moving at v o Source moving at v s 40m/s 20m/s rest 300Hz What do the car driver and the person hear?