Faculty of Computers and Information Fayoum University 5

Faculty of Computers and Information Fayoum University 5 -3 -2018

Chapter 2 1. 2. 3. 4. 5. Speed of Waves on Strings Reflection and Transmission Sinusoidal Waves Rate of Energy Transfer by Sinusoidal Waves Linear wave equation in the string

Objectives: The student will be able to Demonstrate the speed of wave on the string. Define the reflection & transmission waves. The liner equation of the wave. Define the kinetic and the potential energies. Define the total energy on the string Determine the rat of the energy transfer. 5 -3 -2018

Speed of Waves on Strings How do we determine the speed of a transverse pulse traveling on a string? If a string under tension is pulled sideways and released, the tension is responsible for accelerating a particular segment of the string back to the equilibrium position. The acceleration of the So what happens when the tension particular segment increases? increases Which The speed of the wave means? increases. Now what happens when the mass per unit length of the string increases? For the given tension, acceleration decreases, so the wave speed decreases. 5 -3 -2018

Speed of Waves on Strings Which law does this hypothesis based on? Based on the hypothesis we have laid out above, we can construct a hypothetical formula for the speed of wave Is the above expression dimensionally sound? 5 -3 -2018 Newton’s second law of motion T: Tension on the string m: Unit mass per length T=[MLT-2], m=[ML-1] (T/m)1/2=[L 2 T-2]1/2=[LT-1]

Speed of Waves on Strings v q T Ds Fr q q q T Let’s consider a pulse moving to right and look at it in the frame that moves along with the pulse. Since in the reference frame moves with the pulse, the segment is moving to the left O with the speed v, and the centripetal acceleration of the segment is Now what do the force components look in this motion when θ is small? R What is the mass of the segment when the line density of the string is m? Using the radial force component Therefore the speed of the pulse is 5 -3 -2018

Example A uniform cord has a mass of 0. 300 kg and a length of 6. 00 m. The cord passes over a pulley and supports a 2. 00 kg object. Find the speed of a pulse traveling along this cord. Since the speed of wave on a string 1. 00 m with line density m and under the T is M=2. 00 kg tension The line density is The tension on themstring 5. 00 m is provided by the weight of the object. Therefore Thus the speed of the wave is 5 -3 -2018

The speed of a traveling wave • A fixed point on a wave has a constant value of the phase, i. e. Or 5 -3 -2018

The speed of a traveling wave • For a wave traveling in the opposite direction, we simply set time to run backwards, i. e. replace t with -t. • So, general sinusoidal solution is: • In fact, any function of the form 5 -3 -2018 is a solution.

Speed of waves on a string Transverse wave: Speed of wave Where: T… tension in rope (don’t confuse with period T) µ = m/L mass per unit length of rope 5 -3 -2018

Sinusoidal Waves Equation of motion of a simple harmonic oscillation is a sine function. But it does not travel. Now how does wave form look like when the wave travels? The function describing the position of particles, located at x, of the medium through which the sinusoidal wave is traveling can be written at t=0 The wave form of the wave traveling at the speed v in +x at any given time t becomes By definition, the speed Thus the of wave in terms of wave form length and period T is can be rewritten Defining, angular The wave number k form and angular becomes General Wave frequency Frequency, w, wave speed, v f, form 5 -3 -2018 Amplitude Wave Length

Example A sinusoidal wave traveling in the positive x direction has an amplitude of 15. 0 cm, a wavelength of 40. 0 cm, and a frequency of 8. 00 Hz. The vertical displacement of the medium at t=0 and x=0 is also 15. 0 cm. a) Find the angular wave number k, period T, angular frequency w, and speed v of the wave. Using the definition, angular wave number k is Angular Period frequency is is Using period and wave length, the wave speed is b) Determine the phase constant φ, and write a general expression of the wave function. At x=0 and t=0, y=15. 0 cm, therefore the phase f becomes Thus the general wave function is 5 -3 -2018

Sinusoidal Waves on Strings Let’s consider the case where a string is attached to an arm undergoing a simple harmonic oscillation. The trains of waves generated by the motion will travel through the string, causing the particles in the string to undergo simple harmonic motion on y-axis. What does this If the wave at mean? t=0 is The wave function can be written This wave function describes the vertical motion of any point on the string at any time t. Therefore, we can use this function to obtain transverse speed, vy, and acceleration, ay. These are the speed and acceleration of the particle in the medium not of the wave. The maximum speed and How do these look the acceleration of the for simple harmonic particle in the medium at motion? position x at time t are 5 -3 -2018

Example A string is driven at a frequency of 5. 00 Hz. The amplitude of the motion is 12. 0 cm, and the wave speed is 20. 0 m/s. Determine the angular frequency w and angular wave number k for this wave, and write and expression for the wave function. Using frequency, the angular frequency is Angular wave number k is Thus the general expression of the wave function is 5 -3 -2018

Quiz: Suppose you create a pulse by moving the free end of a taut string up and down once with your hand beginning at t =0. The string is attached at its other end to a distant wall. The pulse reaches the wall at time t. Which of the following actions, taken by itself, decreases the time interval that it takes for the pulse to reach the wall? More than one choice may be correct. (a) moving your hand more quickly, but still only up and down once by the same amount (b) moving your hand more slowly, but still only up and down once by the same amount (c) moving your hand a greater distance up and down in the same amount of time (d) moving your hand a lesser distance up and down in the same amount of time (e) using a heavier string of the same length and under the same tension (f) using a lighter string of the same length and under the same tension (g) using a string of the same linear mass density but under decreased tension (h) using a 5 -3 -2018 of the same linear mass density but under increased tension. string

Answer Quiz: • Only answers (f) and (h) are correct. • (a) and (b) affect the transverse speed of a particle of the string, but not the wave speed along the string. • (c) and (d) change the amplitude. • (e) and (g) increase the time interval by decreasing the wave speed. 5 -3 -2018

Reflection of a traveling wave on rigid wall - If a wave encounters a “denser”, new medium, or a rigid wall, it gets reflected. - In this case the reflected pulse is inverted upon reflection 5 -3 -2018

Reflection of a traveling wave on a loose end - If a wave encounters a “less dense” medium or an end it also gets reflected. - In this case the reflected pulse is not inverted upon reflection. 5 -3 -2018

Transmission: Light string heavier string The transmitted pulse is not inverted. The reflected pulse is inverted. 5 -3 -2018

Transmission: Heavy string light sting The transmitted pulse is not inverted. The reflected pulse is not inverted. 5 -3 -2018

Reflection and Transmission A pulse or a wave undergoes various changes when the medium it travels changes. Depending on how rigid the support is, two radically different reflection patterns can be observed. 1. The support is rigidly fixed: The reflected pulse will be inverted to the original due to the force exerted on to the string by the support in reaction to the force on the support due to the pulse on the string. 2. The support is freely moving: The reflected pulse will maintain the original shape but moving in the reverse If the boundary is intermediate between the above two direction. extremes, part of the pulse reflects, and the other undergoes transmission, passing through the boundary and propagating in the new medium. When a wave pulse travels from medium A to B: • v. A> v. B (or m. A<m. B), the pulse is inverted upon reflection. • v. A< v. B(or m. A>m. B), the pulse is not inverted upon reflection. 5 -3 -2018

Rate of Energy Transfer by Sinusoidal Waves on Strings Waves traveling through medium carries energy. source performs work on the string, the energy When an external enters into the string and propagates through the medium as wave. is the potential energy of one wave length of a What traveling Dx, Dm wave? Elastic potential energy of a particle in a simple harmonic motion Since The energy DU of the segment w 2=k/m Dm is As Dx 0, the energy DU becomes Using the wave function, the energy is For the wave at t=0, the potential energy in one wave length, l, is Recall K = 2 p/l 5 -3 -2018

Rate of Energy Transfer by Sinusoidal Waves How does the kinetic energy of each segment of the string in the wave look? Since the vertical speed of the particle is The kinetic energy, DK, of the segment Dm is As Dx 0, the energy DK becomes For the wave at t=0, the kinetic energy in one wave length, l, is Recall K = 2 pl Just like harmonic oscillation, the total mechanical energy in one wave length, l, is 5 -3 -2018

Rate of Energy Transfer by Sinusoidal Waves As the wave moves along the string, the amount of energy passes by a given point changes during one period. So the power, the rate of energy transfer becomes P of any sinusoidal wave is proportion to the square of angular frequency, the square of amplitude, density of medium, and wave speed. 5 -3 -2018

Example A taut string for which m=5. 00 x 10 -2 kg/m is under a tension of 80. 0 N. How much power must be supplied to the string to generate sinusoidal waves at a frequency of 60. 0 Hz and an amplitude of 6. 00 cm? The speed of the wave is Using the frequency, angular frequency w is Since the rate of energy transfer is 5 -3 -2018

The Linear Wave Equation If the wave function has the form This is the linear wave equation as it applies to waves on a string. 5 -3 -2018