Waves and Oscillations Summary Notes

• Finding the equilibrium length of a spring with a mass attached.

  • Apply Hooke’s law and balance the force due to the spring with the gravitational force on the mass. For example, in question 1.1, set the force exerted by the spring ($k(y-l_0)$) equal to the gravitational force ($mg$) and solve for $y$.

• Deriving the equation of motion for a mass on a spring.

  • Use the conservation of mechanical energy principle and differentiate it with respect to time. In question 1.2, this involves differentiating the sum of kinetic and potential energy of the system.

• Verifying a solution for the motion of a mass on a spring.

  • Substitute the given solution into the differential equation derived in the previous step and show that it satisfies the equation. For question 1.3, substitute $y(t)$ into the equation of motion and verify it satisfies the equation using the provided condition on ${\omega }_0$.

• Determining a constant in a solution given an initial condition.

  • Apply the initial condition to the provided solution and solve for the unknown constant. In question 1.4, set $y(0)=l_0$ and solve for $A$.

• Analyzing forces on a pendulum and deriving its energy expression.

  • Draw a free-body diagram and use it to write expressions for kinetic and potential energy. For question 2.1 and 2.2, identify gravitational and tension forces and express mechanical energy in terms of $\theta$ and $\dot{\theta }$.

• Deriving the equation of motion for a pendulum.

  • Differentiate the energy expression with respect to time and apply small angle approximations to simplify. In question 2.3, use the mechanical energy derivative and the approximation $\sin (\theta )\approx \theta$.

• Verifying a solution for the motion of a pendulum.

  • Substitute the proposed solution into the simplified equation of motion and verify it satisfies the equation. In question 2.4, check if $\theta (t)$ is a solution given the condition on ${\omega }_0$.

• Analyzing the effect of acceleration on a pendulum in a car.

  • Apply Newton’s laws in a non-inertial reference frame and solve for the conditions given. In question 3.1 and 3.2, use the force balance in the presence of acceleration and find $\theta$ or $a_x$ accordingly.

• Deriving the wave equation for a string and its solutions.

  • Use the general wave equation and boundary conditions to find the solution form. In questions 4.1 and 4.2, derive the wave equation and express the general solution as a sum over harmonics.

• Calculating frequencies for harmonics in a string.

  • Relate the properties of the string to the frequencies of its harmonics. In question 4.3 and 4.4, use the wave equation to find ${\omega }_n$ and calculate specific frequencies given string properties.

• Deriving the Doppler effect for sound.

  • Use the Doppler effect formula to find the observed frequency for a moving source. In questions 5.1 and 5.2, apply the Doppler effect formula for the source moving towards and away from the observer.

• Calculating observed frequencies in Doppler effect scenarios.

  • Substitute the given values into the Doppler effect formula. In question 5.3, calculate the observed frequencies using the provided values of $v_s$, $c$, and $f$.