1. 1 Animations from: Wikipedia and http://zonalandeducation.com/mstm/physics/waves/partsOfAWave/waveParts.htm#pictureOfAWave WAVES Antonio J. Barbero, Mariano Hernández, Alfonso Calera, Pablo Muñiz, José A. de Toro and Peter Normile Dpt. of Applied Physics. UCLM

2. 2 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). Vibration PropagationVibration Propagation A kind of transverse waves can propagate in the vacuum (electromagnetic waves). However, longitudinal waves can only propagate in a material medium.

3. 3 INTRODUCTORY MATH OF WAVES Wave equation Sign + Waveform traveling to the right Waveform traveling to the left Sign - SpaceTime Phase velocity X Y X Y Waveform f 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)

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

5. 5 Time dependence for x = x 0 t y Wave profile for t = t 0 y x HARMONIC WAVES / 2 Harmonic wave equation (choosing cosine form) Phase velocity Space Time Remember: cosine is periodic. Periodic function is that which verifies See that harmonic waves have double periodicity Period Phase Amplitude Initial phase Displacement space time Trough Crest A -A Same phase points Wavelength Period Snapshot graphHistory graph

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

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

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

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

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

11. 11 VELOCITY OF MECHANICAL WAVES Mechanical waves need a material medium to propagate. Its velocity of propagation depends upon the properties of the medium. Fluids   density of the fluid (kg/m 3 ) Compressibility modulus Solids   density of the solid (kg/m 3 ) Young modulus String   linear density of the string (kg/m) VELOCITY AND ACCELERATION OF THE PARTICLES OF THE MEDIUM Maximum velocity Maximum acceleration

12. 12 WAVES CARRY ENERGY Every section of the string (mass  m) moves up and down because the energy carried by the wave. 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 From the wave equation we obtain for the element  m in the fixed position x 0 Taking into account that k. x 0 is constant, this can be rewritten as This is the equation of the harmonic motion described by the mass element  m. The angular frequency of that motion is . Let us remind that the energy of the mass  m in a harmonic motion (angular frequency , amplitude A) is given by Maximum velocity Let  be the mass per unit of lenght  x of the string Power transmitted by the wave Units: Joule/second = watt

13. 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 (  ): When the traveling wave (towards the right) is reflected at the end, its phase changes  radians (it is inverted). Reflected wave, direction (  ): 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. 14 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: 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! 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. STANDING WAVES / 2 For a given lenght L, the standing waves appears only when the frequencies satisfy that condition. From the relationship among frequency and wavelength (f = v/, where v is the propagation velocity) Velocity is given by n = 1  f 1 fundamental frequency n > 1  f n higher harmonics Node Anti-node This exemple: 4 th harmonics n = 4 n+1 nodes n antinodes

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

16. 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 4 th harmonic of a standing wave. The mass of the string is 5  10 -3 kg/m. 4 th harmonic means n = 4  from L = n /2 we obtain a) Find the tension of the string 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 f 4 = 4 f 1 )