Wave on String flash cards

1. 1.What is a mechanical wave?

   A disturbance that propagates through a material medium, transferring energy and momentum without any net bulk transport of the medium itself. It requires a medium with inertia and elasticity.

   Hint: Energy moves, matter oscillates in place.

2. 2.Distinguish a transverse wave from a longitudinal wave.

   In a transverse wave particles oscillate perpendicular to the direction of propagation (e.g. waves on a string). In a longitudinal wave particles oscillate parallel to the propagation direction, forming compressions and rarefactions (e.g. sound in air).

   Hint: Perpendicular vs parallel to travel.

3. 3.Can transverse mechanical waves travel through gases and liquids (bulk)?

   No. Transverse mechanical waves need a medium that supports shear (rigidity). Gases and non-viscous liquids cannot sustain shear stress, so transverse mechanical waves do not propagate through their bulk; only through solids and on liquid surfaces.

   Hint: Shear modulus needed.

4. 4.Sound in air is what type of wave?

   A longitudinal wave: air particles oscillate back and forth along the direction of propagation creating alternating compressions (high pressure) and rarefactions (low pressure).

   Hint: Compressions and rarefactions.

5. 5.Write the general equation of a wave travelling in the $+x$ direction.

   $y(x,t)=A\sin(kx-\omega t+\phi)$, where $A$ is amplitude, $k$ the angular wavenumber, $\omega$ the angular frequency and $\phi$ the phase constant.

   Hint: $kx-\omega t$ for +x travel.

6. 6.For $y=A\sin(kx-\omega t)$, what does a wave moving in the $-x$ direction look like?

   $y=A\sin(kx+\omega t)$. A relative $+$ sign between the $kx$ and $\omega t$ terms indicates propagation in the negative $x$ direction.

   Hint: $kx+\omega t$.

7. 7.Define wavelength $\lambda$ and its relation to wavenumber $k$.

   Wavelength is the spatial period — the distance over which the wave pattern repeats. It relates to the angular wavenumber by $k=\dfrac{2\pi}{\lambda}$.

   Hint: $k=2\pi/\lambda$.

8. 8.Relate angular frequency $\omega$, frequency $f$, and time period $T$.

   $\omega=2\pi f=\dfrac{2\pi}{T}$, and $f=\dfrac{1}{T}$. Frequency is oscillations per second (Hz); period is seconds per oscillation.

   Hint: $\omega=2\pi/T$.

9. 9.State the fundamental wave relation between speed, frequency and wavelength.

   $v=f\lambda=\dfrac{\omega}{k}$. The wave speed equals the product of frequency and wavelength.

   Hint: $v=f\lambda$.

10. 10.Is wave speed $v$ determined by the source or by the medium?

    For mechanical waves the speed is set by the medium's properties (elasticity and inertia). The source fixes the frequency; then $\lambda=v/f$ adjusts accordingly.

    Hint: Medium sets $v$, source sets $f$.

11. 11.When a wave passes from one medium to another, what stays constant?

    The frequency stays the same (it is fixed by the source). Speed and wavelength both change together so that $v=f\lambda$ holds in each medium.

    Hint: $f$ is invariant across a boundary.

12. 12.Define the phase of a wave $y=A\sin(kx-\omega t)$.

    The phase is the argument $(kx-\omega t)$. It determines the state of oscillation (displacement) of a particle at position $x$ and time $t$.

    Hint: The whole angle inside sine.

13. 13.What is the phase difference between two points separated by $\Delta x$ at one instant?

    $\Delta\phi=k\,\Delta x=\dfrac{2\pi}{\lambda}\Delta x$. A path difference of $\lambda$ corresponds to a phase difference of $2\pi$.

    Hint: $\Delta\phi=(2\pi/\lambda)\Delta x$.

14. 14.Relate path difference and phase difference.

    $\dfrac{\text{path difference}}{\lambda}=\dfrac{\text{phase difference}}{2\pi}$, i.e. $\Delta x=\dfrac{\lambda}{2\pi}\Delta\phi$.

    Hint: Ratio equals $\lambda:2\pi$.

15. 15.Give the expression for particle velocity in a wave $y=A\sin(kx-\omega t)$.

    $v_p=\dfrac{\partial y}{\partial t}=-A\omega\cos(kx-\omega t)$. Its maximum magnitude is $A\omega$. This is different from the wave (phase) speed $v=\omega/k$.

    Hint: $\partial y/\partial t$, max $=A\omega$.

16. 16.How is particle velocity related to the slope of the waveform?

    $v_p=-v\,\dfrac{\partial y}{\partial x}$, i.e. particle velocity $=-(\text{wave speed})\times(\text{slope})$. Particle velocity is largest where the slope is steepest (at $y=0$).

    Hint: $v_p=-v\times$ slope.

17. 17.Give the particle acceleration for $y=A\sin(kx-\omega t)$.

    $a_p=\dfrac{\partial^2 y}{\partial t^2}=-\omega^2 y$. Each particle executes SHM with acceleration proportional to and opposite its displacement.

    Hint: $a=-\omega^2 y$ (SHM).

18. 18.Write the differential (linear) wave equation.

    $\dfrac{\partial^2 y}{\partial x^2}=\dfrac{1}{v^2}\dfrac{\partial^2 y}{\partial t^2}$. Any function of the form $f(x\mp vt)$ satisfies it, with $v$ the wave speed.

    Hint: Second derivatives, factor $1/v^2$.

19. 19.Give the speed of a transverse wave on a stretched string.

    $v=\sqrt{\dfrac{T}{\mu}}$, where $T$ is the tension and $\mu$ is the linear mass density (mass per unit length).

    Hint: $\sqrt{T/\mu}$.

20. 20.For a string, express $\mu$ in terms of density and radius.

    $\mu=\dfrac{m}{L}=\rho A=\rho(\pi r^2)$, where $\rho$ is volume density, $A$ cross-sectional area and $r$ radius. So $v=\sqrt{T/(\rho\pi r^2)}$.

    Hint: $\mu=\rho\pi r^2$.

21. 21.If tension in a string is quadrupled, how does wave speed change?

    Since $v\propto\sqrt{T}$, quadrupling $T$ multiplies $v$ by $\sqrt{4}=2$. The speed doubles.

    Hint: $v\propto\sqrt{T}$.

22. 22.How does wave speed on a string depend on its thickness for the same material and tension?

    $v=\sqrt{T/\mu}$ with $\mu\propto r^2$, so a thicker string (larger $r$) has larger $\mu$ and hence smaller speed. Speed $\propto 1/r$.

    Hint: Thicker = heavier per length = slower.

23. 23.Give the average power transmitted by a wave on a string.

    $P=\tfrac{1}{2}\mu\omega^2 A^2 v$. Power carried is proportional to the square of amplitude and to the square of frequency.

    Hint: $P\propto A^2\omega^2$.

24. 24.What is the intensity of a wave and how does it depend on amplitude?

    Intensity is average power per unit area, $I=\dfrac{P}{\text{Area}}$. For a given medium and frequency, $I\propto A^2$ (and $I\propto\omega^2$).

    Hint: $I\propto A^2$.

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