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WAVES. Antonio J. Barbero, Mariano Hernández, Alfonso Calera, Pablo Muñiz, José A. de Toro and Peter Normile Dpt. of Applied Physics. UCLM. Animations from: Wikipedia and Vibration. Propagation.

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    1. WAVES Antonio J. Barbero, Mariano Hernández, Alfonso Calera, Pablo Muñiz, José A. de Toro and Peter Normile Dpt. of Applied Physics. UCLM Animations from: Wikipedia and

    2. Vibration Propagation Vibration Propagation A wave is a periodic disturbance in space and time, able to propagate energy. The wave equation describes mathematically how the disturbance proceeds across the space and over time. Transverse waves: The oscillations occur perpendicularly to the direction of energy transfer. Exemple: a wave in a tense string. Here the varying magnitude is the distance from the equilibrium horizontal position. Longitudinal waves: Those in which the direction of vibration is the same as their direction of propagation. So the movement of the particles of the medium is either in the same or in the opposite direction to the motion of the wave. Exemple: sound waves, what changes in this case is the pressure of the medium (air, water or whatever it be). A kind of transverse waves can propagate in the vacuum (electromagnetic waves). However, longitudinal waves can only propagate in a material medium.

    3. Time Space Y Phase velocity Sign - Y X X INTRODUCTORY MATH OF WAVES The wave equation describes a traveling wave if the group (x  vt) is present. This is a necessary condition. (The term traveling wave is used to emphasize that we refer here to waves propagating in the medium, not to standing waves that we will consider later) Wave equation Waveform f Waveform traveling to the right Sign + Waveform traveling to the left Waveform f

    4. y depends only upon the time HARMONIC WAVES ? A wave is said to be harmonic when its waveform f is either a sine or a cosine function Harmonic wave moving to the right One more stuff: Whenever a harmonic wave propagates through a medium, every point in the medium describes a harmonic motion or  is a distance Wave equation We can choose any of them by adding an initial phase 0 into the argument of the function… …what physically means that we choose the initial time upon our convenience For exemple: If the wave reaches a maximum for t = 0 and we choose as a reference the cosine waveform, we have that 0 = 0 and the wave equation becomes simply What do we have to do to write the same waveform by using the sine form? Answer: That describes exactly the same wave Wave profile for t = 0 Remember:

    5. Period Time Space Wavelength Same phase points Crest y y A x t -A Trough Period Wave profile for t = t0 Time dependence for x = x0 HARMONIC WAVES / 2 Harmonic wave equation (choosing cosine form) Remember: cosine is periodic. Periodic function is that which verifies Phase Displacement Initial phase Amplitude Phase velocity space See that harmonic waves have double periodicity time Snapshot graph History graph

    6. Time Space 2nd wave 1st wave 3rd wave HARMONIC WAVES / 3 Harmonic wave equation (choosing cosine form) Phase Displacement Displacement: current value of the magnitude y, depending upon space and time. Its maximum value is the amplitude A. Initial phase Wavelength : distance between two consecutive points whose difference of phase is 2. Amplitude Phase velocity Wavenumber k: is the number of waves contained into a turn (2 radians). Sometimes it is called angular or circular wavenumber. Period T: time elapsed till the phase of the harmonic wave increases 2 radians. Its units (I.S.) are rad/m, but often they are referred as m-1. Frequency f: is the inverse of the period, so the frequency tells us the number of oscillations per unit of time. Its units (I.S.) are s-1 (1 s-1 = 1 Hz). Angular requency : is the number of oscillations in a phase interval of 2 radians. Phase velocity is given by the quotient In terms of wavenuber and angular frequency the harmonic wave equation can be written as

    7. Wave equation y (m) Each of those profiles indicates the shape of the pulse for the given time. x (m) SOME EXAMPLES Example 1: traveling pulse This pulse moves to the right (positive direction of X axis) with a velocity of 0.50 m/s where x, y are in meter, t in seconds, v = 0.50 m/s Let us to plot y for different values of time t = 10 t = 5 t = 0

    8. y (m) Each of those profiles indicates the shape of the pulse for the given time. x (m) SOME EXAMPLES / 2 Exemple 2: traveling pulse Wave equation This pulse moves to the left (negative direction of X axis) with a velocity of 0.50 m/s. See that vt = t/2. where x, y are in meter, t in seconds Let us to write the wave equation in such a way that the group x+v·t appears explicitly. Plotting for different values of time t = 0 t = 2 t = 4

    9. y (m) x (m) SOME EXAMPLES / 3 This wave moves to the right (positive direction of X axis) with a velocity of 1.00 m/s Exemple 3: harmonic traveling wave Harmonic wave where x, y are in meter, t in seconds Compare with t = 0 t = 1 t = 2

    10. SOME EXAMPLES / 4 Exemple 4 Harmonic wave This wave moves to the right (positive direction of X axis) with a velocity of 0.50 m/s where x, y are in meter, t in seconds Wavenumber and angular frequency y (m) Comparing A = 1 m, and Phase velocity x (m)

    11. VELOCITY OF MECHANICAL WAVES Mechanical waves need a material medium to propagate. Its velocity of propagation depends upon the properties of the medium. Compressibility modulus   density of the fluid (kg/m3) Fluids Young modulus   density of the solid (kg/m3) Solids   linear density of the string (kg/m) String VELOCITY AND ACCELERATION OF THE PARTICLES OF THE MEDIUM Maximum velocity Maximum acceleration

    12. Maximum velocity WAVES CARRY ENERGY Let us consider a transverse wave in a tensestring. We’ll see that as the wave passes through, every point of the string describes a harmonic motion Every section of the string (mass Dm) moves up and down because the energy carried by the wave. Taking into account that k.x0 is constant, this can be rewritten as From the wave equation we obtain for the element Dm in the fixed position x0 This is the equation of the harmonic motion described by the mass element Dm. The angular frequency of that motion is w. Let us remind that the energy of the mass Dm in a harmonic motion (angular frequency w, amplitude A) is given by Power transmitted by the wave Let m be the mass per unit of lenght Dx of the string Units: Joule/second = watt

    13. STANDING WAVES A standing wave is the result of the superposition of two harmonic wave motions of equal amplitude and equal frequency which propagate in opposite directions through a medium. However the standing wave IS NOT a traveling wave, since its equation does not contain terms of the form (k x - t). For simplicity, we will take as an example to illustrate the formation of standing waves a transverse wave that propagates towards the right () on a string attached at its ends. This wave, reflected on the right end, arises a new wave propagating in the left direction () Incident wave, direction (): Reflected wave, direction (): When the traveling wave (towards the right) is reflected at the end, its phase changes  radians (it is inverted). Every point of the string vibrates with harmonic motion of amplitude 2A sen kx: see that the amplitude depens upon the position, but the group kx-t does not appear. This is to say, the result is not a traveling wave.

    14. From the relationship among frequency and wavelength (f = v/, where v is the propagation velocity) STANDING WAVES / 2 Does any pair of incident and reflected waves arise standing waves in a string, does not matter which the frequency or the wavenumber are? NO! As the ends of the string are fixed, the vibration amplitude at those points must be zero. If we call L the length of the string, at any time the following conditions must be verified: The equation L = n/2 means that standing waves only appear when the length L of the string is an integer multiple of a half-wavelength. For a given lenght L, the standing waves appears only when the frequencies satisfy that condition. Velocity is given by n = 1  f1 fundamental frequency Node Node Node Node Node This exemple: 4th harmonics n = 4 n+1 nodes n antinodes n > 1  fn higher harmonics Anti-node Anti-node Anti-node Anti-node

    15. STANDING WAVES / 3 A standing wave on a string n = 1  f1 fundamental frequency 7th HARMONIC n = 2  f2 2nd harmonic n = 3  f3 3rd harmonic Weights to tense the string

    16. STANDING WAVES / EXEMPLE Two traveling waves of 40 Hz propagate in opposite directions along a 3 m-lenght tense string given rise to the 4th harmonic of a standing wave. The mass of the string is 510-3 kg/m. a) Find the tension of the string 4th harmonic means n = 4  from L = n/2 we obtain b) The amplitude of the antinodes is 3.25 cm. Write the equation of this harmonic of the standing wave c) Find the fundamental frequency for this tense string. The velocity of propagation is constant, and we have the fundamental frequency when (All harmonics are integer multiples of the fundamental frequency, so f4 = 4 f1)