The University of Sydney

School of Mathematics and Statistics

Solutions to Video Assignment 1

MATH3078: PDEs and Waves Semester 2, 2020

Web Page: https://canvas.sydney.edu.au/courses/27747

Lecturer: Daniel Hauer, Tutor: Timothy A. Collier

PDE and Waves

Directions

Please record a video of 3 minutes or less with your mobile phone, webcam, or tablet explaining how

to solve the given problem below. Please upload this video on the Canvas page of this unit of study.

While it may be possible to identify students from the video, please do not put your name on your

assignment so that we can apply anonymous marking as far as possible.

The problem

1. (MATH3078 only)

(a) (Transport equation of water in 1D). We want to derive the PDE modeling the

height of water in a gently-sloping river that is assumed to have a constant slope and

vertical banks. Let h = h(x, t) denote the height of the water surface at position x and

time t and v = v(x, t) the flow velocity. The flow function φ = φ(x, t) satisfies the linear

constitutive law φ = v h.

We further assume that rain (or small tributaries) add water and seepage into the ground

remove water. This will increase the depth of water at any point in the river at rate

r = r(h, x, t). By starting from the classic conservation law, derive the transport equation

ht + (hv)x = r. (1)

(b) (Inviscid Burgers’ equation in 1D). Next, we want to derive the PDE modeling the

flow of water in a gently-sloping river that is assumed to have a constant slope and vertical

banks. We keep the notion introduced in (a). Then, in the simplest model the resistive force

R increases in direct proportion to the flow velocity v, so that R = a v, for some a ∈ R, and

the component of gravitational force F downstream increases in proportion to the depth of

water h, giving F = bh, for some b ∈ R. The flow speed v adjusts itself such that these two

forces R and F are in balance, giving av = bh, or simply

v = κh for κ = b/a.

Use this together with transport equation (1) to derive the inviscid Burgers’ equation

ut + uux = f

for the appropriate scaled functions u of v and f of r.

2. (MATH3978/MATH4078)

(a) Derive the PDE of the (2-dimensional wave equation) modeling the vibration in vertical

direction of a fixed membrane by following the description in [1, Section 12.8].

(b) Explain with reference to your proof of part (a) where the mass conservation law is used.

Copyright© 2020 The University of Sydney 1

Marking criteria of the video assignment

Content (6 marks in total).

• 2 marks. Accurate presentation of mathematical ideas at the correct level of this unit.

• 2 marks. Flow of ideas – is the presentation in a logical order; is the content of the presentation

synthesized into a coherent whole.

• 2 marks. Evidence of understanding of the mathematics.

Presentation-style (4 marks in total).

• 2 marks. Verbal presentation is the delivery clear, confident and aurally interesting.

• 2 marks. Visual presentation (life written, prepared parts) are the visuals clear, well-designed

and confidently handled; does the visual presentation enhance the content of the presentation?

References

[1] E. Kreyszig, Advanced engineering mathematics, John Wiley & Sons, Inc., New York, tenth ed.,

2011.

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