**Oscillations, Waves, and Fluids**

April 27 2018

- oscillations are repetitive motion
- oscillations of many-particle systems lead to wave motion
**Oscillatory Motion**
- oscillatory motion is universal
- systems in stable equilibrium naturally tend to return toward equilibrium when displaced
- remarkably, the mathematical description of oscillatory motion is also universal
- amplitude is the maximum displacement from equilibrium
- period is the time it takes for the motion to repeat itself
- frequency is the number of cycles per time, f = 1/T
- simple harmonic motion
- the type of motion resulting from a restoring force proportional to displacement
- this is often a very good approximation to the real world
- F = -kx
- position in simple harmonic motion is a sinusoidal function of time
- x(t) = A cos wt
- where w is angular frequency
- an angular quantity that provides the simplest mathematical description of the motion
- since circular motion results from perpendicular simple harmonic motions
- so we use the term angular frequency even though there are no angles involved
- w = 2 pi f = 2 pi / T
- w = sqrt (k / m)
**Wave Motion**
- a wave is a travelling disturbance that transports energy but not matter
- mechanical waves
- disturbances of some material medium
- a disturbance in one part of a medium is communicated to adjacent parts
- matter undergoes localized oscillatory motion but does not travel with the wave
- longitudinal wave - local oscillations are in the same direction as the wave propagation
- transverse wave - local oscillations are perpendicular to the wave propagation
- waves travel at specific speeds through its medium
- wavelength is the distance over which the wave pattern repeats
- wave speed v = wavelength / period = wavelength x frequency
- a travelling sinusoidal wave has the form
- y(x, t) = A cos (kx +/- wt)
- where for each constant t, the wave is a snapshot sinusoidal wave
- and where for each constant x, the y-direction exhibits simple harmonic motion
- k is the wave number, which we can find by holding time constant
- k = 2 pi / wavelength
- while w = 2 pi / period is a measure of time frequency, cycles per unit time
- k is a measure of spatial frequency, cycles per unit distance
- the 2 pi again helps make the math simpler
- the relations between k, wavelength, w, and T allow us to write wave speed
- wave speed = wavelength / period = w / k
**Fluid Motion**
- fluid is matter that flows under the influence of external forces
- includes both liquids and gases
- density is mass per unit volume
- pressure is force per unit area
- at a given point in a fluid, pressure is exerted equally in all directions
- hydrostatic equilibrium
- for a fluid to remain at rest, the net force everywhere must be zero
- since pressure is exerted equally in all directions in a fluid
- pressure differences, rather than pressure itself, gives rise to forces within fluids
- equilibrium in the presence of gravity requires a pressure force to counteract
- since forces arise only in pressure differences
- fluid pressure must therefore vary with depth
- dp/dh = density x g
- density of liquids is constant since they are essentially incompressible
- density of gases vary with height since they are compressible
- buoyancy
- Archimedes' principle
- the buoyancy force on an object is equal to the weight of the displaced fluid
- fluid dynamics
- the principles are based on the same Newtonian principles
- but we use macroscopic properties to make the analysis simpler
- in steady flow, the pattern of fluid motion remains the same at each point
- conservation of mass suggestions for any fluid
- density x velocity x area is a constant
- for liquids with constant density
- it means velocity x area is a constant
- the fluid flows faster across smaller cross sections
- conservation of energy, Bernoulli's equation suggests
- the total energy per unit volume of fluid is conserved as the fluid moves
- pressure + 1/2 density v2 + density x g x y is a constant
- smaller cross section
- results in increased velocity
- and lower pressure