الحلقة الخامس التداخل في الموجات الرنين الباب السابع عشر الموجات جدول أعمال حلقة اليوم: 1- مراجعة سريعة و استطراد 2- إنهاء الباب السابع عشر 3- حل مسائل 4- ماذا حققنا من أهداف الدورة 5- خاتمة

17.4 Wavelength and Frequency: (cont’d) If the disturbance is sinusoidal, then: y(x,t) = y m sin(k x –  t +  ) Let’s use Mathematica to try to understand the concepts of amplitude, wavelength, [angular] wavenumber, period, frequency, angular frequency, and phase. waves

17.5 The Speed of a Traveling Wave: Let’s see how fast the wave travels; e.g., how fast the crest moves. k x –  t +  = a constant v =  /k v = /T v = f The wave moves one wavelength per period!! The speed of the wave (v) is dx/dt; therefore,

17.6 Wave Speed on a Stretched String: Use dimensional analysis and/ or see the proof on page 379 A string with linear mass density , under a tension  has a speed: What happens to the speed when the frequency increases?

Example: The wave on a string in the figure below, drawn at time t = 0, is moving to the right. 1.Determine the amplitude, wavelength, angular wavenumber and phase angle (for the sin wave). 2.You are told that the string has a LMD of 40  g/cm and is under 10 N of tension, find the period and angular frequency of the wave. 3.Find the general mathematical expression describing the wave. 4.How would this change of the wave were moving to the left? Mathematica output

17.7 Energy and Power of a Traveling String Wave: Where is the kinetic energy minimum/maximum? Where is the elastic potential energy minimum/maximum? [see the proof on page 381-2] P avg = ½  v  2 y 2 m

17.7 Energy and Power of a Traveling String Wave: (cont’d) In exams, we play games with the students. For example, what happens if we increase the tension on the string by a factor of 9? P avg = ½  v  2 y 2 m The average power transmitted in the wave depends on the linear mass density, on the speed, on the square of the frequency and on the square of the amplitude. Interaction question: What happens if we increase the wavelength by a factor of 10, keeping the tension constant?

17.8 The Principle of Superposition of Waves: y res (x,t) = y 1 (x,t) + y 2 (x,t) Two (or more) overlapping waves algebraically add to produce a resultant (or, net) wave. The overlapping waves do not alter the motion of each other. Let’s see this superposition Mathematica codesuperposition

17.9 Interference of Waves: Two waves propagating along the same direction with the same amplitude, wavelength and frequency, but differing in phase angle will interfere with each other in a nice way. Let’s see the (same) waves Mathematica codewaves Checkpoint #5:

17.11 Standing Waves: What happens when two waves propagating in opposite directions with the same amplitude, wavelength and frequency will interfere with each other such as to create standing waves!! Let’s see the (same) waves Mathematica codewaves

17.11 Standing Waves: (cont’d) Reflection at a Boundary: 1- Hard Reflection 2- Soft Reflection You’re going to love Mathematica; see this code. Reflection-Transmission Checkpoint #6:

17.12 Standing Waves and Resonance: When a string of length L is clamped between two points, and sinusoidal waves are sent along the string, there will be many reflections off the clamped ends. At specific frequencies, interference will produce nodes and large anti-nodes. We say we are at resonance, and that the string is resonating at resonant frequencies. Let’s see the (same) waves Mathematica codewaves

17.12 Standing Waves and Resonance: (cont’d) Example: A 75 cm long string has a wave speed of 10 m/s, and is vibrating in its third harmonic. Find the distance between two adjacent anti-nodes. The fundamental mode (n=1) has a fundamental frequency: f 1 = v/(2L) The second harmonic (n=2) has a frequency: f 2 = 2 f 1 = v/(L) The n th harmonic has a frequency: f n = n f 1 = nv/(2L) The wavelength of the n th harmonic is: n = 2L/n What is the distance between two adjacent nodes (or anti-nodes)? What is the distance between a node and its neighboring anti-node?

Two waves are described as follows: y 1 (x,t) = 4 (x - v*t) y 2 (x,t) = 4 (x + v*t) At what position and time do these two waves cancel? A1 At x = 0 and at any time t. A2 At x = 0 and at t = 0 only. A3 They never cancel (they always add up). A4 At t = 0 and at any position x. A5 They always cancel because v has opposite signs.

A sinusoidal wave is described as: y = (0.1 m) * sin[10*pi*(x/5 + t - 3/2)], where x is in meters and t is in seconds. What are the values of its frequency (f), and its velocity (v)? A1 f=5 Hz, v = 5 m/s moving in -x-direction. A2 f=5 Hz, v = 5 m/s moving in +x-direction. A3 f=2 Hz, v = 1 m/s moving in -x-direction. A4 f=2 Hz, v = 1 m/s moving in +x-direction. A5 f=2 Hz, v = 5 m/s moving in -x-direction.

A transverse harmonic wave in a string is described by: y(x,t) = (3.0 m) * sin(0.3 x - 8 t - Phi), where x is in meters and t is in seconds. At t = 0 and x = 0, a point on the string has a positive displacement and has velocity of 0. The phase constant (Phi) is: A1 270 degrees. A2 180 degrees. A3 135 degrees. A4 90 degrees. A5 45 degrees.

الهدف من الدورة التعرف على أساسيـّات الحركة الاهتزازية و الموجات بما في ذلك الصوت. تعلم "بعض" المواضيع المتقدمة "نوعا ما" مثل الرنين، و الإيقاع و غير ذلك. عرض بعض برامج مثاماتيكا المتعلقة بهذا الموضوع. التدرب على حل مسائل في الحركة الاهتزازية و الموجات. تطوير ثقافة التعلـّم و التعليم عند المتعلـّم. التعوّد على القراءة في المراجع الأساسية باللغة الإنجليزية.