Mechanical Waves and Sound
Students model wave behavior and apply frequency, wavelength, amplitude, and wave speed to mechanical waves and sound phenomena.
Illustrations are auto-generated and may be placeholders. They can be refreshed to match the narration.
Wave Pulses and Media
A mechanical wave is a disturbance that transfers energy through matter without carrying the matter along with it. A single disturbance is called a wave pulse. In a transverse pulse, particles of the medium move perpendicular to the direction the pulse travels. A pulse moving along a stretched rope is an example: the pulse travels horizontally while each piece of rope moves up and down. In a longitudinal pulse, particles move parallel to the wave’s direction, producing compressions and rarefactions. Sound in air is longitudinal. As a pulse passes, particles oscillate around their equilibrium positions rather than traveling with the pulse. Wave behavior depends on the medium. For example, increasing the tension in a rope generally increases the pulse’s speed because the restoring force is greater.
Amplitude and Energy
Amplitude is the maximum displacement of a medium from its equilibrium position. On a displacement-versus-position graph, amplitude is measured vertically from the equilibrium line to a crest or trough, not from crest to trough. A wave with greater amplitude causes particles in the medium to move farther from equilibrium and carries more energy. For many ideal mechanical waves, the energy transported is proportional to the square of the amplitude. Therefore, doubling the amplitude produces four times as much transported energy, if other conditions remain unchanged. For example, shaking one end of a rope 3 centimeters above and below equilibrium creates a larger-amplitude wave than shaking it only 1 centimeter. The larger wave is not necessarily faster; its speed is mainly determined by properties of the medium, such as tension and mass per unit length.
Frequency and Wavelength
Frequency, wavelength, and period describe a repeating wave. Frequency f is the number of complete cycles passing a point each second and is measured in hertz, where 1 Hz equals 1 cycle per second. The period T is the time for one cycle, so f = 1/T. Wavelength λ is the distance between corresponding points on consecutive cycles, such as crest to crest or compression to compression. It is measured in meters. On a position graph, wavelength is a horizontal distance; on a time graph, the horizontal interval between crests is the period. For example, if 12 crests pass a point in 3.0 seconds, the frequency is 4.0 Hz and the period is 0.25 second. If adjacent crests are 0.75 meter apart, the wavelength is 0.75 meter.
The Wave-Speed Equation
Wave speed connects frequency and wavelength through the equation v = fλ, where v is wave speed in meters per second, f is frequency in hertz, and λ is wavelength in meters. The equation follows from the fact that one wavelength passes a point during one period. In a given medium under fixed conditions, wave speed is usually constant. If the source frequency increases, the wavelength decreases so that the product fλ remains equal to the same speed. For example, a wave on a rope has a frequency of 5.0 Hz and a wavelength of 2.4 meters. Its speed is v = (5.0 Hz)(2.4 m) = 12 m/s. If the frequency is then increased to 8.0 Hz without changing the rope or its tension, the wavelength becomes λ = 12 m/s ÷ 8.0 Hz = 1.5 meters.
Interference and Standing Waves
When two or more waves overlap, their displacements add according to the principle of superposition. Constructive interference occurs when displacements in the same direction combine to produce a larger amplitude. Destructive interference occurs when opposite displacements reduce or cancel one another. After overlapping, wave pulses continue traveling with their original shapes in an ideal medium. Continuous waves moving in opposite directions can form a standing wave. Nodes are points that remain at zero displacement, while antinodes oscillate with maximum amplitude. A string fixed at both ends must have a node at each end. Its allowed wavelengths satisfy λn = 2L/n, where L is string length and n is a positive integer. For example, a 0.60-meter string vibrating in its second harmonic has λ2 = 2(0.60 m)/2 = 0.60 meter and has one internal node.
Sound and Resonance
Sound is a longitudinal mechanical wave made of pressure variations traveling through a medium. It cannot travel through a vacuum. In air, vibrating objects create alternating compressions and rarefactions. Frequency mainly determines perceived pitch, while amplitude and intensity relate to perceived loudness. Resonance occurs when a periodic force drives a system near one of its natural frequencies, producing a large-amplitude response. Air columns resonate only at wavelengths that fit their boundary conditions. In a tube closed at one end and open at the other, the fundamental pattern has a displacement node at the closed end and an antinode at the open end, so L = λ/4. For example, if a closed tube is 0.85 meter long, then λ = 3.4 meters. Using 340 m/s for sound speed, its fundamental frequency is f = v/λ = 100 Hz.