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# 2: Waves

In this chapter, we will study the physics of wave motion. We concentrate on mechanical waves, which are disturbances that move through a medium such as air or water. Like simple harmonic motion studied in the preceding chapter, the energy transferred through the medium is proportional to the amplitude squared. The concepts presented in this chapter will be the foundation for many interesting topics, from the transmission of information to the concepts of quantum mechanics.

• 2.1: Prelude to Wave
Surface water waves in the ocean are transverse waves in which the energy of the wave travels horizontally while the water oscillates up and down due to some restoring force. A buoy can be used to convert the awesome power of ocean waves into electricity. The up-and-down motion of the buoy generated as the waves pass is converted into rotational motion that turns a rotor in an electric generator.
• 2.2: Simple Harmonic Motion
A very common type of periodic motion is called simple harmonic motion (SHM). A system that oscillates with SHM is called a simple harmonic oscillator. In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.
• 2.3: Energy in Simple Harmonic Motion
The simplest type of oscillations are related to systems that can be described by Hooke’s law, F = −kx, where F is the restoring force, x is the displacement from equilibrium or deformation, and k is the force constant of the system. Elastic potential energy stored in the deformation of a system can be described by Hooke’s law as U = (1/2)kx^2. Energy in the simple harmonic oscillator is shared between elastic potential energy and kinetic energy, with the total being constant.
• 2.4: Comparing Simple Harmonic Motion and Circular Motion
A projection of uniform circular motion undergoes simple harmonic oscillation. Consider a circle with a radius A, moving at a constant angular speed ω. A point on the edge of the circle moves at a constant tangential speed of v_max = Aω. The projection of the radius onto the x-axis is x(t) = Acos(ωt + ϕ), where (ϕ) is the phase shift. The x-component of the tangential velocity is v(t) = −Aωsin(ωt + ϕ).
• 2.5: Traveling Waves
A wave is a disturbance that moves from the point of origin with a wave velocity v. Mechanical waves are disturbances that move through a medium and are governed by Newton’s laws. Electromagnetic waves are disturbances in the electric and magnetic fields, and do not require a medium. A transverse wave has a disturbance perpendicular to the wave’s direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.
• 2.6: Mathematics of Waves
A wave is an oscillation that travels through a medium, accompanied by a transfer of energy. Energy transfers from one point to another in the direction of the wave motion. The particles of the medium oscillate up and down, back and forth, or both, around an equilibrium position. Given a function of a wave that is a snapshot of the wave, and is only a function of the position x, the motion of the pulse or wave moving at a constant velocity can be modeled by replacing x with x ∓ vt.
• 2.7: Wave Speed on a Stretched String
The speed of a wave on a string depends on the linear density of the string and the tension in the string. The linear density is mass per unit length of the string. In general, the speed of a wave depends on the square root of the ratio of the elastic property to the inertial property of the medium. The speed of sound through air at T = 20 °C is approximately v_s = 343.00 m/s.
• 2.8: Energy and Power of a Wave
The energy and power of a wave are proportional to the square of the amplitude of the wave and the square of the angular frequency of the wave. Intensity is defined as the power divided by the area. As the wave moves out from a source, the energy is conserved, but the intensity decreases as the area increases.
• 2.9: Interference of Waves
Superposition is the combination of two waves at the same location. Constructive interference occurs from the superposition of two identical waves that are in phase. Destructive interference occurs from the superposition of two identical waves that are 180° out of phase. The wave that results from the superposition of two sine waves that differ only by a phase shift is a wave with an amplitude that depends on the value of the phase difference.
• 2.10: Standing Waves and Resonance
A standing wave is the superposition of two waves which produces a wave that varies in amplitude but does not propagate. Nodes are points of no motion in standing waves. An antinode is the location of maximum amplitude of a standing wave. Normal modes of a wave on a string are the possible standing wave patterns. The lowest frequency that will produce a standing wave is known as the fundamental frequency. The higher frequencies which produce standing waves are called overtones.
• 2.E: Waves (Exercises)
• 2.S: Waves (Summary)

Thumbnail: Surfer at Mavericks, one of the world's premier big wave surfing locations. (Surfer: Andrew Davis). (CC SA-BY 2.0; Shalom Jacobovitz).

• Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).