Chapter 5: Waves and periodic motion (C7869116)

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1 Periodic motion

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Periodic motion is motion repeated in regular intervals known as periods. Frequency is the number of occurrences of periodic motion per unit time.

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What are characteristics of waves?

2 Wave characteristics

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Wave is an oscillation that travels through space over time. They are unique because energy is transferred, but there is no permanent displacement of particles. There are three types of waves, including:

  • Mechanical waves, which is a wave that requires a medium to propagate. There are two types of mechanical waves, including:
    • Transverse waves, where vibration is perpendicular to the direction the wave is propagating. For example, a vibrating string is a transverse wave
    • Longitudinal waves, where vibration is parallel and antiparallel to the direction the wave is propagating. For example, a sound wave is a longitudinal wave
  • Electromagnetic waves, which can propagate through a vacuum,
  • Matter waves (aka de Broglie waves), which is the related wave of any particle, as all waves exhibit wave-particle duality. It is defined by [latex]\lambda=\dfrac{h}{p}[/latex], where [latex]h[/latex] is Planck’s constant

Basic waves can be mathematically described by the equation [latex]u(x,t)=A.sin(kx-\omega t+\phi)[/latex], where wavenumber [latex]k=\dfrac{2\pi}{\lambda}[/latex] (discussed ), and angular frequency [latex]\omega=2\pi f[/latex] . Notable characteristics of the wave include:

  • Wavelength (lambda, [latex]\lambda[/latex]), the distance over which the wave repeats [from peak to peak, or trough to trough], and has the SI unit meters
  • Frequency ([latex]f[/latex]), the number of wavelengths that have been repeated, over one second, and has the units cycles per second, or Hertz
  • Period ([latex]T[/latex]), the reciprocal ([latex]\dfrac{1}{x}[/latex]) of frequency [latex]T=\dfrac{1}{f}[/latex], and is the time required for an entire wavelength to cycle, and has the units seconds
  • Amplitude ([latex]A[/latex]), the maximum oscillation of the wave
  • Phase ([latex]\phi[/latex]), the lateral shift. Note that because of the nature of the sine function, [latex]2\pi=360^{\circ}[/latex] which represents an entire wavelength. Thus, the shift of half a wavelength is [latex]\pi=180^{\circ}[/latex]

Water waves are surface waves, which are mechanical waves propagating along the interface of different media, in this case, water and air. In shallow water, where the wavelength is much greater than the depth, the velocity of the wave can be approximated as [latex]v_{shallow}=\sqrt{gd}[/latex], where [latex]g[/latex] is the gravitational constant, and [latex]d[/latex] is the depth of water. Thus, in shallow water, velocity is proportional to depth. In deep water, where the wavelength is much smaller than the depth, the velocity of the wave is [latex]v_{deep}=\sqrt{\dfrac{g\lambda}{2\pi}}[/latex]. Thus, in deep water, velocity is proportional to wavelength. Note that since velocity is dependent on wavelength [and therefore frequency], that deep water is a dispersive medium.

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to displacement, [latex]F=-kx[/latex]. Using calculus, it can be found that acceleration is [latex]a=-\omega^2 x[/latex], meaning that acceleration is directly proportional to displacement. Mechanical energy in simple harmonic motion is conserved, as it perfectly converts potential energy to kinetic energy, and vice versa. Examples of simple harmonic motion include a mass on a string, a mass on a pendulum (when there is a small angle). For a mass on the spring, the period is [latex]T=2\pi \sqrt{\dfrac{m}{k}}[/latex], where [latex]m[/latex] is the mass attached, and [latex]k[/latex] is the spring constant. For the pendulum, the period is [latex]T=2\pi \sqrt{\dfrac{l}{g}}[/latex], where [latex]l[/latex] is the length of the string, and [latex]g[/latex] is gravitational acceleration. Note that the period of the pendulum is independent of the mass or amplitude (i.e. in this case, how high it swings).

The wave velocity is defined as [latex]v=f\lambda[/latex] in a non-dispersive medium. Dispersion is when velocity depends on frequency, thereby causing waves of different frequencies to travel at different speeds. In contrast, in non-dispersive medium, all parts of the wave (in spite of frequency) travel at the same speed. An example of dispersion is the separation of white light into its components shown in the rainbow. Note therefore, that in a non-dispersive medium,  is constant for a given medium. Rewriting the formula as [latex]c=f\lambda[/latex], to emphasize the velocity of the wave is constant , the speed of light. Thus, it is [latex]f[/latex] and [latex]\lambda[/latex] that are inversely proportional. Velocity is thus dependent on the medium’s physical properties, which includes the elastic and inertial components, in accordance with [latex]v=\sqrt{\dfrac{elastic}{inertial}}[/latex]. Elasticity will be defined as the property of materials to return to its original shape after deformation. Inertia was  defined as an object’s tendency to remain at its present state of motion. Whereas the elasticity component of a medium contributes to increase velocity, the inertial component will contributes to decrease velocity. For example, in a vibrating string, the velocity is defined as [latex]v=\sqrt{\dfrac{T}{\mu}}[/latex], where [latex]T[/latex] is tension of the string (the elastic component), and [latex]\mu[/latex] is the linear density of the string [latex]\mu=\dfrac{m}{L}[/latex] (the inertial component).

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What is periodic motion?

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Physics - Pre-med science - MR. SHUM'S CLASSROOM