EJC Physics H211 Waves 2023 1. Notes (full)

Ocean waves exhibit both transverse and longitudinal behaviour. It is observed that particles near the water surface move in a vertical circular motion without significant net displacements in their average positions as a wave propagates. This wave motion “flattens” out as the ocean wave reaches shallower waters, resulting in more of a forward/backward ebbing motion near coasts. This phenomenon also partially explains why tsunamis are devastating: as a tsunami reaches shallower water, its speed of propagation is forced to slow down, its horizontal wavelength is forced to decrease and the kinetic energy is converted to gravitational potential energy – the wave height increases dramatically before smashing onto coastal areas. Content • Progressive waves • Transverse and longitudinal waves • Polarisation • Determination of frequency and wavelength of sound waves Learning Objectives: Candidates should be able to: (a) show an understanding of and use the terms displacement, amplitude, period, frequency, phase difference, wavelength and speed (b) deduce, from the definitions of speed, frequency and wavelength, the equation 𝑣 = 𝑓𝜆 (c) recall and use the equation 𝑣 = 𝑓𝜆 (d) show an understanding that energy is transferred due to a progressive wave (e) recall and use the relationship, intensity ∝ (amplitude)2 (f) show an understanding of and apply the concept that a wave from a point source and travelling without loss of energy obeys an inverse square law to solve problems (g) analyse and interpret graphical representations of transverse and longitudinal waves (h) show an understanding that polarisation is a phenomenon associated with transverse waves (i) recall and use Malus’ law ( intensity ∝ cos2 θ) to calculate the amplitude and intensity of a plane polarised electromagnetic wave after transmission through a polarising filter (j) determine the frequency of sound using a calibrated oscilloscope (k) determine the wavelength of sound using stationary waves (* to be done in H212 Superposition)

We think of waves as a “bulk phenomenon” – it happens on a larger scale and are made up of adjacent smaller units (such as water molecules, gas molecules, rope segments) individually exhibiting oscillations. Waves can result in energy (but not matter) being moved from one point to another OR energy can also be confined to a region in space without transfer. In this topic, w e will focus on the former : progressive waves. In our A-Levels journey we will encounter 3 main types of waves: Mechanical waves e.g. water waves, sound waves and seismic waves (earthquakes). They obey Newton’s laws of motion, and can only exist within a material medium (correspondingly liquid water, air, rock). Electromagnetic waves do not need a material medium to exist. They travel through vacuum at a speed of 8110 30 m s0c. −= . Visible light ranges from 400 nm to 700 nm. Take note of the order of magnitude of EM wavelengths: Matter waves is a concept from Quantum Physics in which small particles (e.g. electrons, protons, neutrons or even some molecules) can behave like waves. We will visit it near the end of JC2, in H219. Before continuing it is useful to introduce wavefronts. A wavefront is an imaginary line that joins all the points on a wave that are at the same phase. It helps to visualize how a wave propagates spatially. y direction of wave travel rope 10-12 10-10 10-8 10-6 10-4 10-2 1 102  / m gamma X-ray UV visible IR microwave radio electric field c In progressive waves, energy is propagated from one place to another in the direction of wave travel without bulk movement of medium. adjacent wavefronts

Let’s compare some terms against those introduced under H210 Oscillations: describe oscillation describe waves displacement x distance in a specified direction from equilibrium position of oscillating mass distance in a specified direction from equilibrium position of particle or point on the wave period T time taken for one complete oscillation of the oscillating mass time between adjacent wavefronts frequency f number of complete to-and-fro motions made by the oscillating mass per unit time 1f T= number of wavefronts passing a point per unit time Two types of graphical representations are commonly used to describe waves: variation of displacement with distance variation of displacement with time Shows displacements of all the particles / segments in a wave and how they vary with distance from the source at a particular instant in time. Since it is a spatial freeze -frame of the wave, it is like a photograph of the oscillations. Shows displacement of a single particle / segment and how it varies with time. Since it tracks the displacement of 1 single entity only, it is like a log of displacements. If the wave is travelling to the right, a particle at the origin  is currently moving downwards. This particular particle is currently mo ving upwards. amplitude x0 0 wavelength  distance (from source) crest trough displacement x each particle or segment is oscillating amplitude x0 0 period T time displacement x

v : speed of wave (m s-1) f : frequency of wave (s-1)  : wavelength of wave (m) The speed of a progressive wave can also be seen from the speed of the wavefront. The speed of oscillation of any individual particle or wave segment 22 0v x x=  − is different from the speed of wave profile vf = . Example 1 Use the definitions of wavelength and frequency to deduce the relationship between , f and v, the speed of a wave. Solution Frequency is the number of wavefronts passing a point per unit time. wavelength is the minimum distance between two points with the same phase ; in other words: wavelength is also the distance moved by a wavefront in the time for a complete oscillation of source distance moved by wavefront in a period one time periodv T f = == Example 2 A wave is represented by the graphs shown. Find the speed of the wave. Solution ( ) ( ) 1 1 1 0402 2 00 m s T . f .. v   − = = = = The speed of a progressive wave is the speed at which energy is transferred. The wavelength  of a wave is the minimum distance between two points with the same phase. vf =

Example 3 A sound wave travels through a gas. The gas molecules oscillate with simple harmonic motion. Each gas molecule is of mass 2610 kg53 . − , vibrates at a frequency of 835 Hz, and has an vibration amplitude of 60 nm. The variation with time t of the vibrational kinetic energy EK of a molecule is shown below. Find the (i) period of oscillations, (ii) time interval ( )21t t− , and (iii) maximum speed vmax for one vibrating molecule. (iv) By reference to the speed of sound in a gas at room temperature, comment on vmax. Solution (i) 1 83 1period 0 00120 s 5T. f= = = (ii) time interval ( )21 2 2 0 000599 1 s Tt f . t = = = − (iii) maximum oscillation speed ( ) ( )( )( ) 2 2 2 max 0 0 0 0 9 1 0 835 610 10 2 2 0 00 0 32 ms x x fx v x x .     − −  − = −  = = = = = (iv) speed of propagation of sound waves in gas at room temperature is about 330 ms-1, which is about 5 orders of magnitude larger than maximum speed of oscillation of a gas molecule. Propagation of sound energy does not involve transfer of medium (mass), so can be greater than the maximum speed of vibration of individual particles in the medium. EK Emax t1 t2 t

Here we revisit the concept of phase first introduced in H210 Oscillations: phase  an angular measure (in either degrees or radians) of the fraction of a cycle completed by the oscillating mass phase difference  measure of how much an oscillation is out of step with another oscillation at the same instant in tim