• Motion
• Newton's Second Law of Motion
• The Spring Equation (Hooke's Law)
• Work, Energy, & Power
• Frequency, Wavelength, & Propagation Speed

Physics Review

To fully understand the various acoustical aspects of sound production, it is generally necessary to use complex mathematical methods such as calculus. However, it is possible to understand a great deal about the physical aspects of sound production with just a few simple concepts.

Motion

1. Distance:
   
   
   • A measure of length between two points.
   • Metric system used almost exclusively in this course.
   • In two- or three-dimensions, a position is specified in terms of distances along each of two or three independent coordinate axes.

2. Speed and Velocity:
   
   
   • Speed provides a measure of distance traveled over a period of time.
   • Velocity specifies both the speed of an object as well as its direction of travel.
   • In one dimension, there is essentially no difference between speed and velocity.
   • Instantaneous velocity is given by $v = \frac{dx}{dt}$, where $x$ is displacement and $t$ is time.

3. Acceleration:
   
   
   • Acceleration is defined as the rate of change of speed.
   • Instantaneous acceleration is given by $a = \frac{dv}{dt} = \frac{d^{2}x}{dt^2}$.

Newton's Second Law of Motion

1. Force:
   
   
   • Force = Mass x Acceleration: $F = m a = m \frac{dv}{dt} = m \frac{d^{2}x}{dt^2}$.
   • The mass of an object is a measure of its opposition to acceleration.
   • Mass and weight are often confused. Weight is the force of gravity on an object. Gravity causes objects to free fall with a constant acceleration (9.8 meters/second$^{2}$ on earth). An object's weight will vary depending on a given gravity. An object will have the exact same mass, however, for any gravity.
   • Force is typically measured in Newtons (kg meters/second$^{2}$).

2. Pressure:
   
   
   • Pressure is defined as the force acting perpendicular to a surface divided by the area of that surface: $p = F_{\bot}/A$.
   • Pressure is a particularly useful quantity to consider when dealing with fluids (liquids and gases), such as air.

The Spring Equation (Hooke's Law)

• Force = Spring Constant x Displacement: $F = k x$.
• The constant $k$ is a measure of the spring's stiffness.
• This linear relationship typically applies only for small displacements.
• The Young's modulus and shear modulus of a material are proportionality constants which relate the resulting deformation to an applied external force.
• For fluids, a similar expression relates a pressure change or stress ($dP$) and a resulting dilation or volume strain $dV/V$, given by a volume change $dV$ with respect to the unperturbed volume $V$, via an elastic bulk modulus $B$: $dP = -B \frac{dV}{V}$.

Work, Energy, & Power

1. Work:
   
   
   • Work = Force $\times$ Distance (for a force which is constant in both magnitude and direction).
   • When an applied force causes an object to move, work is done by the force.
   • Work is typically measured in newton-meters, or joules.

2. Energy:
   
   
   • In order to do work, we must expend energy.
   • Energy comes in many forms though we will mainly be concerned with kinetic (energy of motion) and potential (stored) energy in this course.
   • An object of mass $m$ moving with a velocity $v$ has kinetic energy given by $\mathrm{KE} = \frac{1}{2} m v^{2}$.
   • The same object held at a distance $h$ above the floor has potential energy given by $\mathrm{PE} = m g h$, where $g$ is the acceleration of gravity.
   • If the object then falls to the ground, the work done by gravity would also equal $m g h$. As the object falls, potential energy is converted to kinetic energy. The object's final velocity just before hitting the floor can be determine by equating gain of KE to loss of PE:
     
     $$\frac{1}{2} m v^{2} = m g h \hspace{0.3cm} \Rightarrow \hspa... ... h \hspace{0.3cm} \Rightarrow \hspace{0.3cm} v = \sqrt{2 g h}.$$
     
     

3. Power:
   
   
   • Power = Work / Time.
   • Power relates the rate at which work is done.
   • Power is measured in watts (joules/second).

Frequency, Wavelength, & Propagation Speed

1. Frequency:
   
   
   • Generically, the number of times a specified periodic phenomenon occurs within a specified interval.
   • In acoustic contexts, we are typically concerned with the frequency of vibrations (of various media), which is measured in Hertz (Hz).

2. Wavelength:
   
   
   • Vibrations propagate in the form of waves, which involves the transfer of energy through a media.
   • The distance between successive waves is referred to as the wavelength.

3. Propagation Speed:
   
   
   • Vibrations travel with different speeds, depending on the media..
   • Propagation speed, frequency, and wavelength are related by the relationship: $c = f \lambda$.
   • Sound in air travels with a speed roughly equal to 345 meters / second.
   • Propagation speed is dependent on the density and elasticity of a media. A more ``massy'' material will have a lower propagation speed. A ``stiffer'' material will have a higher speed of propagation.