Show that Eq. (4.2-28), Φ=C r θsinθ, satisfies the biharmonic equation, Eq. (4.2-9).

Name: Problem 10, Chapter 4: Advanced strength and Applied Stress Analysis
Uploaded: 2020-02-17T04:48:37-08:00
Duration: 4 min 44 s
Channel: Monique Rousselle Maynard
Description: Show that Eq. (4.2-28), Φ=C r θsinθ, satisfies the biharmonic equation, Eq. (4.2-9).

step 1 For number 37, you are asked to determine a Cartesian equation for the given polar equation, whi

Show that Eq. (4.2-28), Φ=C r θsinθ, satisfies the...

Key Concepts

Biharmonic Equation

The biharmonic equation involves applying the Laplacian operator twice to a function, resulting in ?^4? = 0. It is central to problems in elasticity and fluid dynamics, where it describes phenomena such as the bending of beams and viscous flows. In this context, showing that a candidate solution satisfies the biharmonic equation means that when the Laplacian operator is applied twice to the function, the resulting expression is identically zero.

Verification of a Candidate Solution

Verifying a candidate solution involves substituting the proposed function into the differential equation and computing all the necessary derivatives to check that the equation is indeed satisfied. This process requires careful differentiation and algebraic manipulation. In this example, one must compute the Laplacian of the provided function and then apply the Laplacian again, ensuring that the final expression vanishes, which confirms that the function is a valid solution to the biharmonic equation.

Differential Operators in Coordinate Systems

When dealing with partial differential equations like the biharmonic equation, it is important to correctly apply differential operators in the appropriate coordinate system, which in this case involves both radial and angular components. Using the correct form of the Laplacian is crucial, especially when functions depend on coordinates such as r and ?, since the derivatives with respect to these variables involve specific terms that arise from the geometry of the coordinate system. This ensures that the verification process is mathematically rigorous.

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