18. Waves & Sound

Wave Functions & Equations of Waves

Periodic Traveling Wave Motion as a Function of x AND t | Doc Physics

Wave Equation

Physics - Mechanics: Mechanical Waves (9 of 21) The Wave Equation

y = (7.5 cm) cos[(1.26 cm−1)x − (44.9 s−1)t]

y = (7.5 cm) cos[(2.51 cm−1)x − (89.8 s−1)t]

y = (7.5 cm) sin[(1.26 cm−1)x − (44.9 s−1)t]

y = (7.5 cm) sin[(2.51 cm−1)x − (89.8 s−1)t]

Write the mathematical representation of the wave graphed in the following two figures.<img src="https://static.studychannel.pearsonprd.tech/courses/physics/thumbnails/images/184a92ee"alt=""/>

The Wave Function

The Mathematical Description Of A Wave

Physics - Mechanics: Mechanical Waves (10 of 21) The Wave Equation in 1-Dimension

<math display=“inline” xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mtext>.3</mtext><mtext>&#x00A0;</mtext><mtext>m/s</mtext><mo rspace="thickmathspace">&#x2009;</mo><mi mathvariant="normal" class="MathML-Unit">in</mi><mtext>&#x00A0;</mtext><mtext>the</mtext><mtext>&#x00A0;</mtext><mtext>positive</mtext><mtext>&#x00A0;</mtext><mi>x</mi><mtext>-direction</mtext></mrow></math>

<math display=“inline” xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mtext>.1</mtext><mtext>&#x00A0;</mtext><mtext>m/s</mtext><mo rspace="thickmathspace">&#x2009;</mo><mi mathvariant="normal" class="MathML-Unit">in</mi><mtext>&#x00A0;</mtext><mtext>the</mtext><mtext>&#x00A0;</mtext><mtext>positive</mtext><mtext>&#x00A0;</mtext><mi>x</mi><mtext>-direction</mtext></mrow></math>

<math display=“inline” xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mtext>.76</mtext><mtext>&#x00A0;</mtext><mtext>m/s</mtext><mo rspace="thickmathspace">&#x2009;</mo><mi mathvariant="normal" class="MathML-Unit">in</mi><mtext>&#x00A0;</mtext><mtext>the</mtext><mtext>&#x00A0;</mtext><mtext>negative</mtext><mtext>&#x00A0;</mtext><mi>x</mi><mtext>-direction</mtext></mrow></math>

<math display=“inline” xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mtext>.3</mtext><mtext>&#x00A0;</mtext><mtext>m/s</mtext><mo rspace="thickmathspace">&#x2009;</mo><mi mathvariant="normal" class="MathML-Unit">in</mi><mtext>&#x00A0;</mtext><mtext>the</mtext><mtext>&#x00A0;</mtext><mtext>negative</mtext><mtext>&#x00A0;</mtext><mi>x</mi><mtext>-direction</mtext></mrow></math>

<math display=“inline” xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mtext>.1</mtext><mtext>&#x00A0;</mtext><mtext>m/s</mtext><mo rspace="thickmathspace">&#x00A0;</mo><mi mathvariant="normal" class="MathML-Unit">in</mi><mtext>&#x00A0;</mtext><mtext>the</mtext><mtext>&#x00A0;</mtext><mtext>negative</mtext><mtext>&#x00A0;</mtext><mi>x</mi><mtext>-direction</mtext></mrow></math>

<math display=“inline” xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mtext>.76</mtext><mtext>&#x00A0;</mtext><mtext>m/s</mtext><mo rspace="thickmathspace">&#x2009;</mo><mi mathvariant="normal" class="MathML-Unit">in</mi><mtext>&#x00A0;</mtext><mtext>the</mtext><mtext>&#x00A0;</mtext><mtext>positive</mtext><mtext>&#x00A0;</mtext><mi>x</mi><mtext>-direction</mtext></mrow></math>

The vertical displacement of a wave moving along the x-axis is given by <math display=“inline”><mrow><mi>y</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mrow><mn>0.045</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">m</mi></mrow></mrow><mo>)</mo></mrow><mi>cos</mi><mrow><mo>(</mo><mrow><mrow><mo>(</mo><mrow><mrow><mn>3.4</mn><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">Hz</mi></mrow></mrow><mo>)</mo></mrow><mi>t</mi><mo>&#x2212;</mo><mrow><mo>(</mo><mrow><mn>4.5</mn><mtext>&#x00A0;</mtext><msup><mrow><mo rspace="thickmathspace">&#x2062;</mo><mi mathvariant="normal" class="MathML-Unit">m</mi></mrow><mrow><mo>&#x2212;</mo><mn>1</mn></mrow></msup></mrow><mo>)</mo></mrow><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math>. What is the wave speed and in what direction is it traveling?

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t)=2.30mm cos[(16.98 rad/m^)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (d) wave speed; (e) direction the wave is traveling;

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t)=2.30mm cos[(16.98 rad/m^)x+(742 rad/s)t]. Being more practical, you measure the rope to have a length of 1.35 m and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

Introduction to Wave Function

vT(7cm, 2s) = 23.3 m/s; vT,max = 157 m/s; aT,max = 7854 m/s2

vT(7cm, 2s) = 23.3 m/s; vT,max = 25 m/s; aT,max = 1250 m/s2

vT(7cm, 2s) = 70.8 m/s; vT,max = 157 m/s; aT,max = 7854 m/s2

vT(7cm, 2s) = 70.8 m/s; vT,max = 157 m/s; aT,max = 49,298 m/s2

vT(7cm, 2s) = 107.2m/s; vT,max = 157 m/s; aT,max = 7854 m/s2

vT(7cm, 2s) = 107.2 m/s; vT,max = 157 m/s; aT,max = 49,298 m/s2

The function for some transverse wave is ? = (0.5 m) sin [(0.8 m−1)x − 2?(50 Hz)t + π/3]. What is the transverse velocity at t=2 s, x=7 cm? What is the maximum transverse speed? The maximum transverse acceleration?

Velocity Of A Wave

Phase Angle

A transverse wave is represented by the following function:y = (18 cm) sin [2π (x/2cm) − t/5s + 1/4)]. What is the phase angle of this wave?

Graphs Of Mathematical Representation of Wave

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