Turbos are naturally quieter than Normally Aspirated engines, your car may not need an extra muffler or resonator that they are sometimes called.

Take a GOOD look at your exhaust system, especially the sides of curved pipes - when I thought that I needed just a tail pipe, I could see that these areas had pinholes and were near the end of their life.

If the tail pipe is straight it is possible that your system is shorter than it need be. Our 740s had curved tail pipes.

When I had a 1956 Ford, I used a lot of scrap pipes to cheaply fix its exhaust system as it rusted out, As an experiment, I made the dual exhausts a foot longer and experimented with cutting an inch off at a time to see how it might effect the system - at one length it was quite noisy and would go Blat, Blat, Blat when I let off fill throttle!

If still available, Volvo's exhaust kit is a great fit and has everything you need, Their clamps are great.

Tips

I got a tailpipe cutter (a large p[ipe cutter with 4 cutting wheels - You only need to swing it just over 90 degrees to cut a pipe. I cut about 6 inches after the front most connector and before and after the pipe over the axle. The 6 inches in the front lets you wiggle it around to free it. The front junction is difficult to take apart, but I have collapsed the pipe by driving a round punch into the junction (cut the pipe just after the junction to make it short if you use this technique)

I mark each pipe with a circle at 1 and 2 inches from each female end - that way, I can see that all junctions are equally inserted into its mating pipe.

I lay the new system beside the old system on the ground to see if they look the same - once I had a new system installed on one of our 850s and it hit the shock absorber because it was bent in the rear - the dealer got another new system (850s are one piece) and they showed me that the other new system was bent as well! They corrected my system by heating it and bending it - they did do a nice job. They contacted Volvo of America.

Take a good look at your system from the rear and both sides - you will notice that if it has been done properly, several segments are straight both vertical and horizontal

Read the FAQ!

Yes, your 780 turbo is supposed to have a rear exhaust muffler. The Volvo parts schematics show it’s the same muffler p/n 3514722 used with the B230FT on other turbos of that vintage for 760, 780, 960, and some early 940 (both turbo and NA), but not 740 and later 940. It’s NLA from Volvo (as are most 700/900 mufflers), but aftermarket equivalents that fit should be available, such as from Starla (Walker). You will have to be careful doing parts searches. The only real differences between the original rear mufflers of that vintage for the B230F, B230FT and B234F was the inlet and outlet pipe diameters being a little larger for the turbo and 16-valve, also whether the over axle piping was a separate piece (used with solid rear axle) or integral to the mufflers (yours is shown as integral, which was used with multi-link suspension). The chassis mounting “ears” should otherwise be the same.

Often times an entire muffler system kit might have been installed which could change the original fitments. Depending on how close to original your exhaust system still is, you should verify pipe diameter, over axle connection and be prepared to use adapters as needed. A good muffler shop should be able to easily deal with it. A shop may well suggest just leaving the straight through pipe unless you have a specific reason, such as emissions inspection, collector status, noise or any slight concerns for the exhaust valves, otherwise it should be of no practical consequence.

BTW, I met my first 780 in early 1989 when I was shopping for a new Volvo. This 780 had just come into the showroom after prep detailing and I was one of the first to sit in it. I was totally smitten, but it was quite a bit out of my price range. Always kind of wished I could have one, but parts for 780s have always been an issue so just as glad I didn’t.
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Dave -still with 940's, prev 740/240/140/120 You'd think I'd have learned by now