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