Wave ON String flash cards

1. 1.What is a transverse wave?

   A wave in which particles of the medium oscillate perpendicular to the direction of wave propagation. Waves on a string are transverse.

   Hint: Direction of particle motion vs propagation.

2. 2.Can a transverse mechanical wave travel through a gas or non-viscous liquid (bulk)?

   No. Transverse mechanical waves need a medium that supports shear stress, so they travel only through solids (and surfaces/strings), not through the bulk of gases or ideal liquids.

   Hint: Shear modulus required.

3. 3.Write the general equation of a wave moving in the $+x$ direction.

   $y(x,t)=A\sin(\omega t-kx+\phi)$, equivalently $y=A\sin(kx-\omega t)$ (differs by sign/phase convention).

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

4. 4.In $y=A\sin(\omega t-kx)$, what does each symbol mean?

   $A$ = amplitude, $\omega$ = angular frequency, $k$ = wave number, $x$ = position, $t$ = time; $(\omega t-kx)$ is the phase.

   Hint: Amplitude, angular freq, wave number.

5. 5.How do you tell the direction of travel from a wave equation?

   If $x$ and $t$ appear in the combination $(\omega t-kx)$ the wave moves in $+x$; if $(\omega t+kx)$ it moves in $-x$. Opposite signs of $x,t$ term ⇒ $+x$.

   Hint: Same sign vs opposite sign.

6. 6.Define wavelength $\lambda$.

   The spatial period: distance between two consecutive points in the same phase (e.g. crest to crest). $k=\dfrac{2\pi}{\lambda}$.

   Hint: Crest to crest distance.

7. 7.Define the wave number $k$ and its units.

   $k=\dfrac{2\pi}{\lambda}$, the angular (radian) wave number; units rad/m. It is the phase change per unit length.

   Hint: $2\pi/\lambda$.

8. 8.Define angular frequency $\omega$ and time period $T$.

   $\omega=\dfrac{2\pi}{T}=2\pi f$; $T$ is the time for one full oscillation of a particle. Units of $\omega$: rad/s.

   Hint: $2\pi/T$.

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

   $v=f\lambda=\dfrac{\omega}{k}$. The wave speed equals the ratio of angular frequency to wave number.

   Hint: $v=\omega/k$.

10. 10.What is the phase of a wave and how does it vary?

    Phase $=(\omega t-kx+\phi)$. It increases with time at a fixed point and decreases with $x$ at fixed time (for $+x$ wave).

    Hint: Argument of the sine.

11. 11.What is phase difference for a path difference $\Delta x$?

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

    Hint: $k\,\Delta x$.

12. 12.What is the phase difference between two particles separated by $\lambda/4$?

    $\Delta\phi=\dfrac{2\pi}{\lambda}\cdot\dfrac{\lambda}{4}=\dfrac{\pi}{2}$ (90°).

    Hint: Fraction of $2\pi$.

13. 13.Distinguish wave velocity from particle velocity.

    Wave velocity $v=\omega/k$ is constant and is the speed of energy/phase transport. Particle velocity $v_p=\partial y/\partial t=A\omega\cos(\omega t-kx)$ oscillates and varies with position/time.

    Hint: One constant, one oscillating.

14. 14.Write the particle velocity for $y=A\sin(\omega t-kx)$.

    $v_p=\dfrac{\partial y}{\partial t}=A\omega\cos(\omega t-kx)$; its maximum is $v_{p,\max}=A\omega$.

    Hint: Differentiate w.r.t. $t$.

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

    $a_p=\dfrac{\partial^2 y}{\partial t^2}=-A\omega^2\sin(\omega t-kx)=-\omega^2 y$; maximum magnitude $A\omega^2$.

    Hint: $-\omega^2 y$, like SHM.

16. 16.Relation between particle velocity, slope of string, and wave velocity.

    $v_p=-v\,\dfrac{\partial y}{\partial x}$. Particle velocity $=-$(wave speed)$\times$(slope of the string).

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

17. 17.What is the slope $\partial y/\partial x$ for $y=A\sin(\omega t-kx)$?

    $\dfrac{\partial y}{\partial x}=-Ak\cos(\omega t-kx)$. Maximum slope magnitude $Ak$.

    Hint: Differentiate w.r.t. $x$.

18. 18.State the linear (1-D) wave equation.

    $\dfrac{\partial^2 y}{\partial t^2}=v^2\,\dfrac{\partial^2 y}{\partial x^2}$, where $v$ is the wave speed. Any $y=f(x\pm vt)$ satisfies it.

    Hint: Second derivatives in $t$ and $x$.

19. 19.How is wave speed read off from the differential wave equation?

    In $\dfrac{\partial^2 y}{\partial t^2}=v^2\dfrac{\partial^2 y}{\partial x^2}$, the coefficient of the space second-derivative is $v^2$; also $v=\omega/k$.

    Hint: Coefficient is $v^2$.

20. 20.Speed of a transverse wave on a stretched string.

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

    Hint: Tension over mass per length.

21. 21.Define linear mass density $\mu$.

    $\mu=\dfrac{m}{L}=\rho A$ (mass per unit length), where $\rho$ is volume density and $A$ the cross-section area. Units kg/m.

    Hint: Mass per unit length.

22. 22.How does wave speed on a string change if tension is quadrupled?

    $v\propto\sqrt{T}$, so quadrupling $T$ doubles $v$.

    Hint: Square-root dependence.

23. 23.How does wave speed depend on the string's thickness (same material)?

    $\mu=\rho A\propto r^2$, and $v=\sqrt{T/\mu}$, so thicker string ⇒ larger $\mu$ ⇒ smaller speed for the same tension.

    Hint: $v\propto 1/\sqrt{\mu}$.

24. 24.When a wave passes from a thin to a thick string (same tension), what changes and what stays constant?

    Frequency $f$ stays constant (set by the source). Speed and wavelength decrease in the denser (thicker) string since $v=\sqrt{T/\mu}$ and $\lambda=v/f$.

    Hint: Frequency is fixed across a boundary.

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